Optical apparatus which uses a virtually imaged phased array to produce chromatic dispersion

ABSTRACT

Various configurations of a virtually imaged phased array (VIPA) generator in combination with a mirror to compensate for chromatic dispersion. A VIPA generator produces a light traveling from the VIPA generator. In some embodiments, a variable curvature mirror is positioned to reflect the light back to the VIPA generator. A rotation axis around which the mirror is rotated and a translation path for the rotation axis are provided, to change the curvature of the mirror where the output light is reflected. In other embodiments, a plurality of mirrors have different surface curvatures. A holder has a rotation axis and holds the plurality of mirrors equidistantly from the rotation axis. The holder is rotatable around the rotation axis to bring a different, respective mirror in position to reflect light produced by a VIPA generator back to the VIPA generator. In other embodiments, a rotating mirror is rotatable about a rotation axis to reflect light produced by a VIPA generator to a respective fixed mirror. In still other embodiments, an off-axis parabolic mirror is rotatable about a rotation axis to reflect light produced by a VIPA generator to a respective fixed mirror.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a divisional of application Ser. No. 10/252,389,filed September 24, 2002 now U.S. Pat. No. 6,786,611; which is adivisional of application Ser. No. 10/021,086, filed Dec. 19, 2001, nowU.S. Pat. No. 6,481,861, issued Nov. 19, 2002; which Is a divisional ofapplication Ser. No. 09/727,450, filed Dec. 4, 2000, now U.S. Pat. No.6,343,866, issued Feb. 5, 2002, which is a continuation-in-part (CIP) ofU.S. application Ser. No. 09/576,541, filed May 23, 2000, now U.S. Pat.No. 6,332,689, issued Dec. 25, 2001, which are incorporated herein byreference.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to an apparatus producing chromaticdispersion, and which can be used to compensate for chromatic dispersionaccumulated in an optical fiber transmission line. More specifically,the present invention relates to an apparatus which uses a virtuallyimaged phased array to produce chromatic dispersion.

2. Description of the Related Art

FIG. 1(A) is a diagram illustrating a conventional fiber opticcommunication system, for transmitting information via light. Referringnow to FIG. 1(A), a transmitter 30 transmits pulses 32 through anoptical fiber 34 to a receiver 36. Unfortunately, chromatic dispersion,also referred to as “wavelength dispersion”, of optical fiber 34degrades the signal quality of the system.

More specifically, as a result of chromatic dispersion, the propagatingspeed of a signal in an optical fiber depends on the wavelength of thesignal. For example, when a pulse with a longer wavelength (for example,a pulse with wavelengths representing a “red” color pulse) travelsfaster than a pulse with a shorter wavelength (for example, a pulse withwavelengths representing a “blue” color pulse), the dispersion istypically referred to as “normal” dispersion. By contrast, when a pulsewith a shorter wavelength (such as a blue color pulse) is faster than apulse with a longer wavelength (such as a red color pulse), thedispersion is typically referred to as “anomalous” dispersion.

Therefore, if pulse 32 consists of red and blue color pulses whenemitted from transmitter 30, pulse 32 will be split as it travelsthrough optical fiber 34 so that a separate red color pulse 38 and ablue color pulse 40 are received by receiver 36 at different times. FIG.1(A) illustrates a case of “normal” dispersion, where a red color pulsetravels faster than a blue color pulse.

As another example of pulse transmission, FIG. 1(B) is a diagramillustrating a pulse 42 having wavelength components continuously fromblue to red, and transmitted by transmitter 30. FIG. 1(C) is a diagramillustrating pulse 42 when arrived at receiver 36. Since the redcomponent and the blue component travel at different speeds, pulse 42 isbroadened in optical fiber 34 and, as illustrated by FIG. 1(C), isdistorted by chromatic dispersion. Such chromatic dispersion is verycommon in fiber optic communication systems, since all pulses include afinite range of wavelengths.

Therefore, for a fiber optic communication system to provide a hightransmission capacity, the fiber optic communication system mustcompensate for chromatic dispersion.

FIG. 2 is a diagram illustrating a fiber optic communication systemhaving an opposite dispersion component to compensate for chromaticdispersion. Referring now to FIG. 2, generally, an opposite dispersioncomponent 44 adds an “opposite” dispersion to a pulse to canceldispersion caused by traveling through optical fiber 34.

There are conventional devices which can be used as opposite dispersioncomponent 44. For example, FIG. 3 is a diagram illustrating a fiberoptic communication system having a dispersion compensation fiber whichhas a special cross-section index profile and thereby acts as anopposite dispersion component, to compensate for chromatic dispersion.Referring now to FIG. 3, a dispersion compensation fiber 46 provides anopposite dispersion to cancel dispersion caused by optical fiber 34.However, a dispersion compensation fiber is expensive to manufacture,and must be relatively long to sufficiently compensate for chromaticdispersion. For example, if optical fiber 34 is 100 km in length, thendispersion compensation fiber 46 should be approximately 20 to 30 km inlength.

FIG. 4 is a diagram illustrating a chirped grating for use as anopposite dispersion component, to compensate for chromatic dispersion.Referring now to FIG. 4, light traveling through an optical fiber andexperiencing chromatic dispersion is provided to an input port 48 of anoptical circulator 50. Circulator 50 provides the light to chirpedgrating 52. Chirped grating 52 reflects the light back towardscirculator 50, with different wavelength components reflected atdifferent distances along chirped grating 52 so that differentwavelength components travel different distances to thereby compensatefor chromatic dispersion. For example, chirped grating 52 can bedesigned so that longer wavelength components are reflected at a fartherdistance along chirped grating 52, and thereby travel a farther distancethan shorter wavelength components. Circulator 50 then provides thelight reflected from chirped grating 52 to an output port 54. Therefore,chirped grating 52 can add opposite dispersion to a pulse.

Unfortunately, a chirped grating has a very narrow bandwidth forreflecting pulses, and therefore cannot provide a wavelength bandsufficient to compensate for light including many wavelengths, such as awavelength division multiplexed light. A number of chirped gratings maybe cascaded for wavelength multiplexed signals, but this results in anexpensive system. Instead, a chirped grating with a circulator, as inFIG. 4, is more suitable for use when a single channel is transmittedthrough a fiber optic communication system.

FIG. 5 is a diagram illustrating a conventional diffraction grating,which can be used in producing chromatic dispersion. Referring now toFIG. 5, a diffraction grating 56 has a grating surface 58. Parallellights 60 having different wavelengths are incident on grating surface58. Lights are reflected at each step of grating surface 58 andinterfere with each other. As a result, lights 62, 64 and 66 havingdifferent wavelengths are output from diffraction grating 56 atdifferent angles. A diffraction grating can be used in a spatial gratingpair arrangement, as discussed in more detail below, to compensate forchromatic dispersion.

More specifically, FIG. 6(A) is a diagram illustrating a spatial gratingpair arrangement for use as an opposite dispersion component, tocompensate for chromatic dispersion. Referring now to FIG. 6(A), light67 is diffracted from a first diffraction grating 68 into a light 69 forshorter wavelength and a light 70 for longer wavelength. These lights 69and 70 are then diffracted by a second diffraction grating 71 intolights propagating in the same direction. As can be seen from FIG. 6(A),wavelength components having different wavelengths travel differentdistances, to add opposite dispersion and thereby compensate forchromatic dispersion. Since longer wavelengths (such as lights 70)travel longer distance than shorter wavelengths (such as lights 69), aspatial grating pair arrangement as illustrated in FIG. 6(A) hasanomalous dispersion.

FIG. 6(B) is a diagram illustrating an additional spatial grating pairarrangement for use as an opposite dispersion component, to compensatefor chromatic dispersion. As illustrated in FIG. 6(B), lenses 72 and 74are positioned between first and second diffraction gratings 68 and 71so that they share one of the focal points. Since longer wavelengths(such as lights 70) travel shorter distance than shorter wavelengths(such as lights 69), a spatial grating pair arrangement as illustratedin FIG. 6(B) has normal dispersion.

A spatial grating pair arrangement as illustrated in FIGS. 6(A) and 6(B)is typically used to control dispersion in a laser resonator. However, apractical spatial grating pair arrangement cannot provide a large enoughdispersion to compensate for the relatively large amount of chromaticdispersion occurring in a fiber optic communication system. Morespecifically, the angular dispersion produced by a diffraction gratingis usually extremely small, and is typically approximately 0.05degrees/nm. Therefore, to compensate for chromatic dispersion occurringin a fiber optic communication system, first and second gratings 68 and71 would have to be separated by very large distances, thereby makingsuch a spatial grating pair arrangement impractical.

SUMMARY OF THE INVENTION

Therefore, it is an object of the present invention to provide anapparatus which produces chromatic dispersion, and which is practicalfor compensating for chromatic dispersion accumulated in an opticalfiber.

Objects of the present invention are achieved by providing an apparatuswhich includes a device described herein and which can be referred to asa “virtually imaged phased array”, “VIPA” or “VIPA generator”.

More specifically, objects of the present invention are achieved byproviding an apparatus which includes (a) a variable curvature mirrorpositioned to reflect light produced by a virtually imaged phased array(VIPA) generator back to the VIPA generator; and (b) a rotation axisaround which the mirror is rotated to change the curvature of the mirrorwhere the output light is reflected.

Objects of the present invention are also achieved by providing anapparatus which includes (a) a virtually imaged phased array (VIPA)generator producing a light traveling from the VIPA generator; (b) avariable curvature mirror positioned to reflect the light back to theVIPA generator; and (c) a rotation axis around which the mirror isrotated to change the curvature of the mirror where the output light isreflected.

Objects of the present invention are further achieved by providing anapparatus which includes (a) a virtually imaged phased array (VIPA)generator receiving an input light at a respective wavelength andproducing a corresponding output light traveling from the VIPA generatorin a direction determined by the wavelength of the input light; (b) avariable curvature mirror positioned to reflect the output light back tothe VIPA generator so that the reflected output light travels throughthe VIPA generator, to thereby provide dispersion compensation to theinput light; and (c) a rotation axis around which the mirror is rotatedto change the curvature of the mirror where the output light isreflected, to thereby vary an amount of dispersion compensation providedto the input light.

Moreover, objects of the present invention are achieved by providing anapparatus which includes (a) a radiation window; (b) first and secondreflecting surfaces in parallel with each other, the first reflectingsurface allowing substantially no light to be transmitted therethroughand being in the same plane as the radiation window, the secondreflecting surface having a reflectivity which causes a portion of lightincident thereon to be transmitted therethrough, where an input light ata respective wavelength travels through the radiation window and isfocused into a line, and the first and second reflecting surfaces arepositioned so that the input light radiates from the line to bereflected a plurality of times between the first and second reflectingsurfaces and thereby cause a plurality of lights to be transmittedthrough the second reflecting surface, the plurality of transmittedlights interfering with each other to produce a collimated output lightwhich travels from the second reflecting surface along a directiondetermined by the wavelength of the input light, and is therebyspecially distinguishable from an output light formed for an input lighthaving a different wavelength; (c) a variable curvature mirrorreflecting the output light back to the second reflecting surface topass through the second reflecting surface and undergo multiplereflection between the first and second reflecting surfaces; and (d) arotation axis around which the mirror is rotated to change the curvatureof the mirror where the output light is reflected.

Objects of the present invention are achieved by providing an apparatuswhich includes (a) a plurality of mirrors having different surfacecurvatures to reflect light; and (b) a holder having a rotation axis andholding the plurality of mirrors equidistantly from the rotation axis,the holder being rotatable around the rotation axis to bring adifferent, respective mirror of the plurality of mirrors in position toreflect light produced by a virtually imaged phased array (VIPA)generator back to the VIPA generator.

Objects of the present invention are achieved by providing an apparatuswhich includes (a) a virtually imaged phased array (VIPA) generatorproducing light; (b) a plurality of mirrors having different surfacecurvatures; and (c) a holder having a rotation axis and holding theplurality of mirrors equidistantly from the rotation axis, the holderbeing rotatable around the rotation axis to bring a different,respective mirror of the plurality of mirrors in position to reflect thelight produced by a VIPA generator back to the VIPA generator.

In addition, objects of the present invention are achieved by providingan apparatus which includes (a) a virtually imaged phased array (VIPA)generator receiving an input light at a respective wavelength andproducing a corresponding output light traveling from the VIPA generatorin a direction determined by the wavelength of the input light; (b) aplurality of mirrors having different surface curvatures; and (c) aholder having a rotation axis and holding the plurality of mirrorsequidistantly from the rotation axis, the holder being rotatable aroundthe rotation axis to bring a different, respective mirror of theplurality of mirrors in position to reflect the output light back to theVIPA generator, to thereby provide dispersion compensation to the inputlight.

Objects of the present invention are also achieved by providing anapparatus which includes (a) a radiation window; (b) first and secondreflecting surfaces in parallel with each other, the first reflectingsurface allowing substantially no light to be transmitted therethroughand being in the same plane as the radiation window, the secondreflecting surface having a reflectivity which causes a portion of lightincident thereon to be transmitted therethrough, where an input light ata respective wavelength travels through the radiation window and isfocused into a line, and the first and second reflecting surfaces arepositioned so that the input light radiates from the line to bereflected a plurality of times between the first and second reflectingsurfaces and thereby cause a plurality of lights to be transmittedthrough the second reflecting surface, the plurality of transmittedlights interfering with each other to produce a collimated output lightwhich travels from the second reflecting surface along a directiondetermined by the wavelength of the input light, and is therebyspecially distinguishable from an output light formed for an input lighthaving a different wavelength; (c) a plurality of mirrors havingdifferent surface curvatures; and (d) a holder having a rotation axisand holding the plurality of mirrors equidistantly from the rotationaxis, the holder being rotatable around the rotation axis to bring adifferent, respective mirror of the plurality of mirrors in position toreflect the output light back to the second reflecting surface to passthrough the second reflecting surface and undergo multiple reflectionbetween the first and second surfaces.

Moreover, objects of the present invention are achieved by providing anapparatus which includes (a) a plurality of fixed mirrors havingdifferent surface curvatures to reflect light; and (b) a rotating mirrorrotatable about a rotation axis to reflect light produced by a virtuallyimaged phased array (VIPA) generator to a respective fixed mirror of theplurality of fixed mirrors, and to reflect the light reflected by therespective fixed mirror back to the VIPA generator.

In addition, objects of the present invention are achieved by providingan apparatus which includes (a) a virtually imaged phased array (VIPA)generator producing light; (b) a plurality of fixed mirrors havingdifferent surface curvatures to reflect light; and (c) a rotating mirrorrotatable about a rotation axis to reflect light produced by the VIPAgenerator to a respective fixed mirror of the plurality of fixedmirrors, and to reflect the light reflected by the respective fixedmirror back to the VIPA generator.

Further, objects of the present invention are achieved by providing anapparatus which includes (a) a virtually imaged phased array (VIPA)generator receiving an input light at a respective wavelength andproducing a corresponding output light traveling from the VIPA generatorin a direction determined by the wavelength of the input light; (b) aplurality of fixed mirrors having different surface curvatures toreflect light; and (c) a rotating mirror rotatable about a rotation axisto reflect the output light traveling from the VIPA generator to arespective fixed mirror of the plurality of fixed mirrors, and toreflect the light reflected by the respective fixed mirror back to theVIPA generator, to thereby provide dispersion compensation to the inputlight.

Also, objects of the present invention are achieved by providing anapparatus which includes (a) a radiation window; (b) first and secondreflecting surfaces in parallel with each other, the first reflectingsurface allowing substantially no light to be transmitted therethroughand being in the same plane as the radiation window, the secondreflecting surface having a reflectivity which causes a portion of lightincident thereon to be transmitted therethrough, where an input light ata respective wavelength travels through the radiation window and isfocused into a line, and the first and second reflecting surfaces arepositioned so that the input light radiates from the line to bereflected a plurality of times between the first and second reflectingsurfaces and thereby cause a plurality of lights to be transmittedthrough the second reflecting surface, the plurality of transmittedlights interfering with each other to produce a collimated output lightwhich travels from the second reflecting surface along a directiondetermined by the wavelength of the input light, and is therebyspecially distinguishable from an output light formed for an input lighthaving a different wavelength; (c) a plurality of fixed mirrors havingdifferent surface curvatures to reflect light; and (d) a rotating mirrorrotatable about a rotation axis to reflect the output light travelingfrom the second reflecting surface to a respective fixed mirror of theplurality of fixed mirrors, and to reflect the light reflected by therespective fixed mirror back to the second reflecting surface to passthrough the second reflecting surface and undergo multiple reflectionbetween the first and second reflecting surfaces.

Objects of the present invention are achieved by providing an apparatuswhich includes (a) a plurality of fixed mirrors having different surfacecurvatures to reflect light; and (b) an off-axis parabolic mirrorrotatable about a rotation axis to reflect light produced by a virtuallyimaged phased array (VIPA) generator to a respective fixed mirror of theplurality of fixed mirrors, and to reflect the light reflected by therespective fixed mirror back to the VIPA generator.

Further, objects of the present invention are achieved by providing anapparatus which includes (a) a virtually imaged phased array (VIPA)generator producing light; (b) a plurality of fixed mirrors havingdifferent surface curvatures to reflect light; and (c) an off-axisparabolic mirror rotatable about a rotation axis to reflect lightproduced by the VIPA generator to a respective fixed mirror of theplurality of fixed mirrors, and to reflect the light reflected by therespective fixed mirror back to the VIPA generator.

In addition, objects of the present invention are achieved by providingan apparatus which includes (a) a virtually imaged phased array (VIPA)generator receiving an input light at a respective wavelength andproducing a corresponding output light traveling from the VIPA generatorin a direction determined by the wavelength of the input light; (b) aplurality of fixed mirrors having different surface curvatures toreflect light; and (c) an off-axis parabolic mirror rotatable about arotation axis to reflect the output light traveling from the VIPAgenerator to a respective fixed mirror of the plurality of fixedmirrors, and to reflect the light reflected by the respective fixedmirror back to the VIPA generator, to thereby provide dispersioncompensation to the input light.

Moreover, objects of the present invention are achieved by providing anapparatus which includes (a) a radiation window; (b) first and secondreflecting surfaces in parallel with each other, the first reflectingsurface allowing substantially no light to be transmitted therethroughand being in the same plane as the radiation window, the secondreflecting surface having a reflectivity which causes a portion of lightincident thereon to be transmitted therethrough, where an input light ata respective wavelength travels through the radiation window and isfocused into a line, and the first and second reflecting surfaces arepositioned so that the input light radiates from the line to bereflected a plurality of times between the first and second reflectingsurfaces and thereby cause a plurality of lights to be transmittedthrough the second reflecting surface, the plurality of transmittedlights interfering with each other to produce a collimated output lightwhich travels from the second reflecting surface along a directiondetermined by the wavelength of the input light, and is therebyspecially distinguishable from an output light formed for an input lighthaving a different wavelength; (c) a plurality of fixed mirrors havingdifferent surface curvatures to reflect light; and (d) an off-axisparabolic mirror rotatable about a rotation axis to reflect the outputlight traveling from the second reflecting surface to a respective fixedmirror of the plurality of fixed mirrors, and to reflect the lightreflected by the respective fixed mirror back to the second reflectingsurface to pass through the second reflecting surface and undergomultiple reflection between the first and second reflecting surfaces.

BRIEF DESCRIPTION OF THE DRAWINGS

These and other objects and advantages of the invention will becomeapparent and more readily appreciated from the following description ofthe preferred embodiments, taken in conjunction with the accompanyingdrawings of which:

FIG. 1(A) (prior art) is a diagram illustrating a conventional fiberoptic communication system.

FIG. 1(B) is a diagram illustrating a pulse before transmission througha fiber in a conventional fiber optic communication system.

FIG. 1(C) is a diagram illustrating a pulse after being transmittedthrough a fiber in a conventional fiber optic communication system.

FIG. 2 (prior art) is a diagram illustrating a fiber optic communicationsystem having an opposite dispersion component to compensate forchromatic dispersion.

FIG. 3 (prior art) is a diagram illustrating a fiber optic communicationsystem having a dispersion compensation fiber as an opposite dispersioncomponent.

FIG. 4 (prior art) is a diagram illustrating a chirped grating for useas an opposite dispersion component, to compensate for chromaticdispersion.

FIG. 5 (prior art) is a diagram illustrating a conventional diffractiongrating.

FIG. 6(A) (prior art) is a diagram illustrating a spatial grating pairarrangement for production of anomalous dispersion.

FIG. 6(B) (prior art) is a diagram illustrating a spatial grating pairarrangement for production of normal dispersion.

FIG. 7 is a diagram illustrating a VIPA.

FIG. 8 is a detailed diagram illustrating the VIPA of FIG. 7.

FIG. 9 is a diagram illustrating a cross-section along lines IX—IX ofthe VIPA illustrated in FIG. 7.

FIG. 10 is a diagram illustrating interference between reflectionsproduced by a VIPA.

FIG. 11 is a diagram illustrating a cross-section along lines IX—IX ofthe VIPA illustrated in FIG. 7, for determining the tilt angle of inputlight.

FIGS. 12(A), 12(B), 12(C) and 12(D) are diagrams illustrating a methodfor producing a VIPA.

FIG. 13 is a diagram illustrating an apparatus which uses a VIPA as anangular dispersion component to produce chromatic dispersion.

FIG. 14 is a more detailed diagram illustrating the operation of theapparatus in FIG. 13.

FIG. 15 is a diagram illustrating various orders of interference of aVIPA.

FIG. 16 is a graph illustrating the chromatic dispersion for severalchannels of a wavelength division multiplexed light.

FIG. 17 is a diagram illustrating different channels of a wavelengthdivision multiplexed light being focused at different points on a mirrorby a VIPA.

FIG. 18 is a diagram illustrating a side view of an apparatus which usesa VIPA to provide variable chromatic dispersion to light.

FIG. 19 is a diagram illustrating a side view of an apparatus which usesa VIPA to provide variable chromatic dispersion to light.

FIGS. 20(A) and 20(B) are diagrams illustrating side views of anapparatus which uses a VIPA to provide chromatic dispersion to light.

FIG. 21 is a graph illustrating the output angle of a luminous flux froma VIPA versus wavelength of the luminous flux.

FIG. 22 is a graph illustrating the angular dispersion of a VIPA versusthe wavelength of a luminous flux.

FIG. 23 is a graph illustrating the effect of different mirror types inan apparatus using a VIPA.

FIG. 24 is a diagram illustrating chromatic dispersion versus wavelengthin an apparatus using a VIPA, for different types of mirrors used in theapparatus.

FIG. 25 is a graph illustrating the effect of a mirror in an apparatuswhich uses a VIPA.

FIG. 26 is a graph illustrating constant chromatic dispersion of anapparatus using a VIPA.

FIG. 27 is a graph illustrating characteristics of different mirrordesigns for an apparatus using a VIPA.

FIGS. 28(A), 28(B), 28(C), 28(D), 28(E) and 28(F) are diagramsillustrating examples of mirrors of an apparatus using a VIPA.

FIG. 29 is a diagram illustrating a cylindrical mirror.

FIG. 30(A) is a graph illustrating chromatic dispersion versuswavelength for one channel of a wavelength division multiplexed light,after undergoing chromatic dispersion compensation with a VIPA with acylindrical mirror.

FIG. 30(B) is a graph illustrating chromatic dispersion versuswavelength for all wavelengths of a wavelength division multiplexedlight, after undergoing chromatic dispersion compensation with a VIPAwith a cylindrical mirror.

FIG. 31(A) is a graph illustrating chromatic dispersion versuswavelength for one channel of a wavelength division multiplexed light,after undergoing chromatic dispersion compensation with a VIPA with amodified cylindrical mirror.

FIG. 31(B) is a graph illustrating chromatic dispersion versuswavelength for all wavelengths of a wavelength division multiplexedlight, after undergoing chromatic dispersion compensation with a VIPAwith a modified cylindrical mirror.

FIG. 32 is a diagram illustrating a top view of an apparatus using aVIPA to provide variable chromatic dispersion to light, according to afurther embodiment of the present invention.

FIGS. 33(A) and 33(B) are diagrams illustrating how a mirror can beformed from a section of a cone, according to an embodiment of thepresent invention.

FIG. 34(A) is a graph illustrating the amount of chromatic dispersionversus wavelength within one channel for different radii of curvature ofa mirror in an apparatus using a VIPA to provide chromatic dispersion,according to an embodiment of the present invention.

FIG. 34(B) is a diagram illustrating radii of curvature of FIG. 34(A),according to an embodiment of the present invention.

FIG. 34(C) is a diagram illustrating modified radii of curvature,according to an embodiment of the present invention.

FIG. 35 is a graph illustrating the chromatic dispersion versuswavelength for different radii of curvature in an apparatus using a VIPAto provide chromatic dispersion, according to an embodiment of thepresent invention.

FIG. 36 is a diagram illustrating various angles in an apparatus whichuses a VIPA, according to an embodiment of the present invention.

FIG. 37 is an additional diagram illustrating angles in an apparatuswhich uses a VIPA, according to an embodiment of the present invention.

FIG. 38 is a diagram illustrating how chromatic dispersion is generatedin an apparatus using a VIPA, according to an embodiment of the presentinvention.

FIGS. 39(A), 39(B) and 39(C) are graphs illustrating mirror curves,according to an embodiment of the present invention.

FIG. 40 is a diagram illustrating a cone for forming a mirror, accordingto an embodiment of the present invention.

FIG. 41 is a diagram illustrating a step shaped mirror surface,according to an embodiment of the present invention.

FIG. 42 is a diagram illustrating a side view of an apparatus using aVIPA to provide chromatic dispersion slope, according to an additionalembodiment of the present invention.

FIG. 43(A) is a graph illustrating the amount of chromatic dispersionfor all wavelengths with the apparatus in FIG. 42 using a cone shapedmirror, according to an embodiment of the present invention.

FIG. 43(B) is a graph illustrating the amount of chromatic dispersionfor all wavelengths with the apparatus in FIG. 42 using a modified coneshaped mirror, according to an embodiment of the present invention.

FIG. 44 is a diagram illustrating the use of a holographic gratingbetween a VIPA and a lens, according to an embodiment of the presentinvention.

FIG. 45 is a diagram illustrating the use of a reflection type gratingbetween a VIPA and a lens, according to an embodiment of the presentinvention.

FIGS. 46 and 47 are diagrams illustrating the use of quarter wave plate,according to embodiments of the present invention.

FIG. 48(A) is a diagram illustrating a side or top view of an apparatuswhich uses a VIPA to provide different chromatic dispersion fordifferent channels, according to a still further embodiment of thepresent invention.

FIG. 48(B) is a graph illustrating chromatic dispersion versuswavelength for the apparatus in FIG. 48(A), according to an embodimentof the present invention.

FIG. 49 is a diagram illustrating a side or top view of an apparatuswhich uses a VIPA to provide different chromatic dispersion fordifferent channels, according to an embodiment of the present invention.

FIG. 50 is a graph illustrating insertion loss in an apparatus whichuses a VIPA to provide chromatic dispersion, according to an embodimentof the present invention.

FIG. 51 is a diagram illustrating different diffraction efficiency atdifferent wavelengths in an apparatus which uses a VIPA to providechromatic dispersion, according to an embodiment of the presentinvention.

FIG. 52 is a diagram illustrating the light intensity of light travelingout of a fiber and into a VIPA, according to an embodiment of thepresent invention.

FIG. 53 is a diagram illustrating a side view of an optical phase maskon an input fiber to produce a double-humped shape far fielddistribution, in an apparatus which uses a VIPA to provide chromaticdispersion, according to an embodiment of the present invention.

FIG. 54 is a diagram illustrating a cross-sectional view along lines54—54 in FIG. 53, according to an embodiment of the present invention.

FIG. 55 is a diagram illustrating a side view of phase masks on a VIPAto provide a double-humped shape far field distribution with respect tolight received inside the VIPA, according to an embodiment of thepresent invention.

FIG. 56 is a diagram illustrating a side view of phase masks on a VIPAto provide a double-humped shape far field distribution with respect tolight received inside the VIPA, according to an additional embodiment ofthe present invention.

FIGS. 57 and 58 are diagrams illustrating a side view of phase masks ona VIPA to provide a double-humped shape far field distribution withrespect to light received inside the VIPA, according to an additionalembodiment of the present invention.

FIG. 59 is a diagram illustrating excessive loss added to a loss curve,according to an embodiment of the present invention.

FIG. 60 is a diagram illustrating the use of an excess loss component toprovide excess loss, according to an embodiment of the presentinvention.

FIG. 61 is a diagram illustrating a side view of a mirror for use with aVIPA to provide chromatic dispersion, according to an embodiment of thepresent invention.

FIG. 62 is a diagram illustrating a front view of a mirror, according toan embodiment of the present invention.

FIGS. 63(A), 63(B) and 63(C) are diagrams illustrating a way to modulateeffective reflectivity in an apparatus using a VIPA, according to anembodiment of the present invention.

FIG. 64 is a diagram illustrating the use of a grating between a VIPAand a lens, according to an embodiment of the present invention.

FIGS. 65, 66 and 67 are diagrams illustrating the use of a VIPA with amovable mirror, according to embodiments of the present invention.

FIGS. 68 and 69 are diagrams illustrating a tunable dispersioncompensator that utilizes a rotatable variably curved mirror, accordingto an embodiment of the present invention.

FIG. 70 is a diagram illustrating an example of a variably curved mirrorfor utilization as part of a tunable dispersion compensator, accordingto an embodiment of the present invention.

FIG. 71 is a diagram illustrating a tunable dispersion compensator thatutilizes a plurality of curved mirrors or mirror segments, according toan embodiment of the present invention.

FIG. 72 is a diagram illustrating a tunable dispersion compensator thatutilizes a plurality of curved mirrors or mirror segments and arotatable flat mirror, according to an embodiment of the presentinvention.

FIG. 73 is a diagram illustrating a tunable dispersion compensator thatutilizes a plurality of curved mirrors or mirror segments and arotatable off-axis parabolic mirror, according to an embodiment of thepresent invention.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

Reference will now be made in detail to the present preferredembodiments of the present invention, examples of which are illustratedin the accompanying drawings, wherein like reference numerals refer tolike elements throughout.

FIG. 7 is a diagram illustrating a virtually imaged phased array (VIPA).Moreover, hereinafter, the terms “virtually imaged phased array,” “VIPA”and “VIPA generator” may be used interchangeably.

Referring now to FIG. 7, a VIPA 76 is preferably made of a plate ofglass. An input light 77 is focused into a line 78 with a lens 80, suchas a semi-cylindrical lens, so that input light 77 travels into VIPA 76.Line 78 is hereinafter referred to as “focal line 78”. Input light 77radially propagates from focal line 78 to be received inside VIPA 76.VIPA 78 then outputs a luminous flux 82 of collimated light, where theoutput angle of luminous flux 82 varies as the wavelength of input light77 changes. For example, when input light 77 is at a wavelength λ1, VIPA76 outputs a luminous flux 82 a at wavelength λ1 in a specificdirection. When input light 77 is at a wavelength λ2, VIPA 76 outputs aluminous flux 82 b at wavelength λ2 in a different direction. Therefore,VIPA 76 produces luminous fluxes 82 a and 82 b which are spatiallydistinguishable from each other.

FIG. 8 is a detailed diagram illustrating VIPA 76. Referring now to FIG.8, VIPA 76 includes a plate 120 made of, for example, glass, and havingreflecting films 122 and 124 thereon. Reflecting film 122 preferably hasa reflectance of approximately 95% or higher, but less than 100%.Reflecting film 124 preferably has a reflectance of approximately 100%.A radiation window 126 is formed on plate 120 and preferably has areflectance of approximately 0% reflectance.

Input light 77 is focused into focal line 78 by lens 80 throughradiation window 126, to undergo multiple reflection between reflectingfilms 122 and 124. Focal line 78 is preferably on the surface of plate120 to which reflecting film 122 is applied. Thus, focal line 78 isessentially line focused onto reflecting film 122 through radiationwindow 126. The width of focal line 78 can be referred to as the “beamwaist” of input light 77 as focused by lens 80. Thus, the embodiment ofthe present invention as illustrated is FIG. 8 focuses the beam waist ofinput light 77 onto the far surface (that is, the surface havingreflecting film 122 thereon) of plate 120. By focusing the beam waist onthe far surface of plate 120, the present embodiment of the presentinvention reduces the possibility of overlap between (i) the area ofradiation window 126 on the surface of plate 120 covered by input light77 as it travels through radiation window 126 (for example, the area “a”illustrated in FIG. 11, discussed in more detail further below), and(ii) the area on reflecting film 124 covered by input light 77 wheninput light 77 is reflected for the first time by reflecting film 124(for example, the area “b” illustrated in FIG. 11, discussed in moredetail further below). It is desirable to reduce such overlap to ensureproper operation of the VIPA.

In FIG. 8, an optical axis 132 of input light 77 has a small tilt angleθ. Upon the first reflection off of reflecting film 122, 5% of the lightpasses through reflecting film 122 and diverges after the beam waist,and 95% of the light is reflected towards reflecting film 124. Afterbeing reflecting by reflecting film 124 for the first time, the lightagain hits reflecting film 122 but is displaced by an amount d. Then, 5%of the light passes through reflecting film 122. In a similar manner, asillustrated in FIG. 8, the light is split into many paths with aconstant separation d. The beam shape in each path forms so that thelight diverges from virtual images 134 of the beam waist. Virtual images134 are located with constant spacing 2t along a line that is normal toplate 120, where t is the thickness of plate 120. The positions of thebeam waists in virtual images 134 are self-aligned, and there is no needto adjust individual positions. The lights diverging from virtual images134 interfere with each other and form collimated light 136 whichpropagates in a direction that changes in accordance with the wavelengthof input light 77.

The spacing of light paths is d=2t Sin θ, and the difference in the pathlengths between adjacent beams is 2t Cos θ. The angular dispersion isproportional to the ratio of these two numbers, which is cot θ. As aresult, a VIPA produces a significantly large angular dispersion.

As easily seen from FIG. 8, the term “virtually imaged phased array”arises from the formation of an array of virtual images 134.

FIG. 9 is a diagram illustrating a cross-section along lines IX—IX ofVIPA 76 illustrated in FIG. 7. Referring now to FIG. 9, plate 120 hasreflecting surfaces 122 and 124 thereon. Reflecting surfaces 122 and 124are in parallel with each other and spaced by the thickness t of plate120. Reflecting surfaces 122 and 124 are typically reflecting filmsdeposited on plate 120. As previously described, reflecting surface 124has a reflectance of approximately 100%, except in radiation window 126,and reflecting surface 122 has a reflectance of approximately 95% orhigher. Therefore, reflecting surface 122 has a transmittance ofapproximately 5% or less so that approximately 5% of less of lightincident on reflecting surface 122 will be transmitted therethrough andapproximately 95% or more of the light will be reflected. Thereflectances of reflecting surfaces 122 and 124 can easily be changed inaccordance with the specific VIPA application. However, generally,reflecting surface 122 should have a reflectance which is less than 100%so that a portion of incident light can be transmitted therethrough.

Reflecting surface 124 has radiation window 126 thereon. Radiationwindow 126 allows light to pass therethrough, and preferably has noreflectance, or a very low reflectance. Radiation window 126 receivesinput light 77 to allow input light 77 to be received between, andreflected between, reflecting surfaces 122 and 124.

Since FIG. 9 represents a cross-section along lines IX—IX in FIG. 7,focal line 78 in FIG. 7 appears as a “point” in FIG. 9. Input light 77then propagates radially from focal line 78. Moreover, as illustrated inFIG. 9, focal line 78 is positioned on reflecting surface 122. Althoughit is not required for focal line 78 to be on reflecting surface 122, ashift in the positioning of focal line 78 may cause small changes in thecharacteristics of VIPA 76.

As illustrated in FIG. 9, input light 77 enters plate 120 through anarea A0 in radiation window 126, where points P0 indicate peripheralpoints of area A0.

Due to the reflectivity of reflecting surface 122, approximately 95% ormore of input light 77 is reflected by reflecting surface 122 and isincident on area A1 of reflecting surface 124. Points P1 indicateperipheral points of area A1. After reflecting off area A1 on reflectingsurface 124, input light 77 travels to reflecting surface 122 and ispartially transmitted through reflecting surface 122 as output lightOut1 defined by rays R1. In this manner, as illustrated in FIG. 9, inputlight 77 experiences multiple reflections between reflecting surfaces122 and 124, wherein each reflection off of reflecting surface 122 alsoresults in a respective output light being transmitted therethrough.Therefore, for example, each time immediately after input light 77reflects off of areas A2, A3 and A4 on reflecting surface 124, inputlight 77 reflects off of reflecting surface 122 to produce output lightsOut2, Out3 and Out4. Points P2 indicate peripheral points of area A2,points P3 indicate peripheral points of area A3, and points P4 indicateperipheral points of area A4. Output light Out2 is defined by rays R2,output light Out3 is defined by rays R3 and output light Out4 is definedby rays R4. Although FIG. 9 only illustrates output lights Out0, Out1,Out2, Out3 and Out4, there will actually be many more output lights,depending on the power on input light 77 and the reflectances ofreflecting surfaces 122 and 124. As will be discussed in more detailfurther below, the output lights interfere with each other to produce aluminous flux having a direction which changes in accordance with thewavelength of input light 77. Therefore, the luminous flux can bedescribed as being a resulting output light formed from the interferenceof output lights Out0, Out1, Out2, Out3 and Out4.

FIG. 10 is a diagram illustrating interference between reflectionsproduced by a VIPA. Referring now to FIG. 10, light traveling from focalline 78 is reflected by reflecting surface 124. As previously described,reflecting surface 124 has a reflectance of approximately 100% and,therefore, functions essentially as a mirror. As a result, output lightOut1 can be optically analyzed as if reflecting surfaces 122 and 124 didnot exist and, instead, output light Out1 was emitted from a focal lineI₁. Similarly, output lights Out2, Out3 and Out4 can be opticallyanalyzed as if they were emitted from focal lines I₁, I₂, I₃ and I₄,respectively. The focal lines I₂, I₃ and I₄ are virtual images of afocal line I₀.

Therefore, as illustrated in FIG. 10, focal line I₁ is a distance 2tfrom focal line I₀, where t equals the distance between reflectingsurfaces 122 and 124. Similarly, each subsequent focal line is adistance 2t from the immediately preceding focal line. Thus, focal lineI₂ is a distance 2t from focal line I₁. Moreover, each subsequentmultiple reflection between reflecting surfaces 122 and 124 produces anoutput light which is weaker in intensity than the previous outputlight. Therefore, output light Out2 is weaker in intensity than outputlight Out1.

As illustrated in FIG. 10, output lights from the focal lines overlapand interfere with each other. More specifically, since focal lines I₁,I₂, I₃ and I₄ are the virtual images of focal line I₀, output lightsOut0, Out1, Out2, Out3 and Out4 have the same optical phase at thepositions of focal lines I₁, I₂, I₃ and I₄. Therefore, interferenceproduces a luminous flux which travels in a specific direction dependingon the wavelength of input light 77.

A VIPA according to the above embodiments of the present invention hasstrengthening conditions which are characteristics of the design of theVIPA. The strengthening conditions increase the interference of theoutput lights so that a luminous flux is formed. The strengtheningconditions of the VIPA are represented by the following Equation (1):2t×cos φ=mλ

where φ indicates the propagation direction of the resulting luminousflux as measured from a line perpendicular to the surface of reflectingsurfaces 122 and 124, λ indicates the wavelength of the input light, tindicates the distance between the reflecting surfaces 122 and 124, andm indicates an integer.

Therefore, if t is constant and m is assigned a specific value, then thepropagation direction φ of the luminous flux formed for input lighthaving wavelength λ can be determined.

More specifically, input light 77 is radially dispersed from focal line78 through a specific angle. Therefore, input light having the samewavelength will be traveling in many different direction from focal line78, to be reflected between reflecting surfaces 122 and 124. Thestrengthening conditions of the VIPA cause light traveling in a specificdirection to be strengthened through interference of the output lightsto form a luminous flux having a direction corresponding to thewavelength of the input light. Light traveling in different directionthan the specific direction required by the strengthening condition willbe weakened by the interference of the output lights.

FIG. 11 is a diagram illustrating a cross-section along lines IX—IX ofthe VIPA illustrated in FIG. 7, showing characteristics of a VIPA fordetermining the angle of incidence, or tilt angle, of input light.

Referring now to FIG. 11, input light 77 is collected by a cylindricallens (not illustrated) and focused at focal line 78. As illustrated inFIG. 11, input light 77 covers an area having a width equal to “a” onradiation window 126. After input light 77 is reflected one time fromreflecting surface 122, input light 77 is incident on reflecting surface124 and covers an area having a width equal to “b” on reflecting surface124. Moreover, as illustrated in FIG. 11, input light 77 travels alongoptical axis 132 which is at a tilt angle θ1 with respect to the normalto reflecting surface 122.

The tilt angle θ1 should be set to prevent input light 77 from travelingout of the plate through radiation window 126 after being reflected thefirst time by reflecting surface 122. In other words, the tilt angle θ1should be set so that input light 77 remains “trapped” betweenreflecting surfaces 122 and 124 and does not escape through radiationwindow 126. Therefore, to prevent input light 77 from traveling out ofthe plate through radiation window 126, the tilt angle θ1 should be setin accordance with the following Equation (2):tilt of optical axis θ1≧(a+b)/4t

Therefore, as illustrated by FIGS. 7–11, a VIPA receives an input lighthaving a respective wavelength within a continuous range of wavelengths.The VIPA causes multiple reflection of the input light to produceself-interference and thereby form an output light. The output light isspatially distinguishable from an output light formed for an input lighthaving any other wavelength within the continuous range of wavelengths.For example, FIG. 9 illustrates an input light 77 which experiencesmultiple reflection between reflecting surfaces 122 and 124. Thismultiple reflection produces a plurality of output lights Out0, Out1,Out2, Out3 and Out 4 which interfere with each other to produce aspatially distinguishable luminous flux for each wavelength of inputlight 77.

“Self-interference” is a term indicating that interference occursbetween a plurality of lights or beams which all originate from the samesource. Therefore, the interference of output lights Out0, Out1, Out2,Out3 and Out4 is referred to as self-interference of input light 77,since output lights Out0, Out1, Out2, Out3 and Out4 all originate fromthe same source (that is, input light 77).

An input light can be at any wavelength within a continuous range ofwavelengths. Thus, the input light is not limited to being a wavelengthwhich is a value chosen from a range of discrete values. In addition,the output light produced for an input light at a specific wavelengthwithin a continuous range of wavelengths is spatially distinguishablefrom an output light which would have been produced if the input lightwas at a different wavelength within the continuous range ofwavelengths. Therefore, as illustrated, for example, in FIG. 7, thetraveling direction (that is, a “spatial characteristic”) of theluminous flux 82 is different when input light 77 is at differentwavelengths within a continuous range of wavelengths.

FIGS. 12(A), 12(B), 12(C) and 12(D) are diagram illustrating a methodfor producing a VIPA.

Referring now to FIG. 12(A), a parallel plate 164 is preferably made ofglass and exhibits excellent parallelism. Reflecting films 166 and 168are formed on both sides of the parallel plate 164 by vacuum deposition,ion spattering or other such methods. One of reflecting films 166 and168 has a reflectance of nearly 100%, and the other reflecting film hasa reflectance of lower than 100%, and preferably higher than 80%.

Referring now to FIG. 12(B), one of reflecting films 166 and 168 ispartially shaved off to form a radiation window 170. In FIG. 12(B),reflecting film 166 is shown as being shaved off so that radiationwindow 170 can be formed on the same surface of parallel plate 164 asreflecting film 166. However, instead, reflecting film 168 can bepartially shaved off so that a radiation window is formed on the samesurface of parallel plate 164 as reflecting film 168. As illustrated bythe various embodiment of the present invention, a radiation window canbe on either side of parallel plate 164.

Shaving off a reflecting film can be performed by an etching process,but a mechanical shaving process can also be used and is less expensive.However, if a reflecting film is mechanically shaved, parallel plate 164should be carefully processed to minimize damage to parallel plate 164.For example, if the portion of parallel plate 164 forming the radiationwindow is severely damaged, parallel plate 164 will generate excess losscaused by scattering of received input light.

Instead of first forming a reflecting film and then shaving it off, aradiation window can be produced by preliminarily masking a portion ofparallel plate 164 corresponding to a radiation window, and thenprotecting this portion from being covered with reflecting film.

Referring now to FIG. 12(C), a transparent adhesive 172 is applied ontoreflecting film 166 and the portion of parallel plate 164 from whichreflecting film 166 has been removed. Transparent adhesive 172 shouldgenerate the smallest possible optical loss since it is also applied tothe portion of parallel plate 164 forming a radiation window.

Referring now to FIG. 12(D), a transparent protector plate 174 isapplied onto transparent adhesive 172 to protect reflecting film 166 andparallel plate 164. Since transparent adhesive 172 is applied to fillthe concave portion generated by removing reflecting film 166,transparent protector plate 174 can be provided in parallel with the topsurface of parallel plate 164.

Similarly, to protect reflecting film 168, an adhesive (not illustrated)can be applied to the top surface of reflecting film 168 and should beprovided with a protector plate (not illustrated). If reflecting film168 has a reflectance of about 100%, and there is no radiation window onthe same surface of parallel plate 164, then an adhesive and protectorplate do not necessarily have to be transparent.

Furthermore, an anti-reflection film 176 can be applied on transparentprotector plate 174. For example, transparent protector plate 174 andradiation window 170 can be covered with anti-reflection film 176.

A focal line can be on the surface of a radiation window or on theopposite surface of a parallel plate from which input light enters.Moreover, the focal line can be in the parallel plate, or before theradiation window.

In accordance with the above, two reflecting films reflect lighttherebetween, with the reflectance of one reflecting film beingapproximately 100%. However, a similar effect can be obtained with tworeflecting films each having a reflectance of less than 100%. Forexample, both reflecting films can have a reflectance of 95%. In thiscase, each reflecting film has light traveling therethrough and causinginterference. As a result, a luminous flux traveling in the directiondepending on the wavelength is formed on both sides of the parallelplate on which the reflecting films are formed. Thus, the variousreflectances of the various embodiments of the present invention caneasily be changed in accordance with required characteristics of a VIPA.

In accordance with the above, a waveguide device is formed by a parallelplate, or by two reflecting surfaces in parallel with each other.However, the plate or reflecting surfaces do not necessarily have to beparallel.

In accordance with the above, a VIPA uses multiple-reflection andmaintains a constant phase difference between interfering lights. As aresult, the characteristics of the VIPA are stable, thereby reducingoptical characteristic changes causes by polarization. By contrast, theoptical characteristics of a conventional diffraction grating experienceundesirable changes in dependance on the polarization of the inputlight.

In accordance with the above, a VIPA provides luminous fluxes which are“spatially distinguishable” from each other. “Spatially distinguishable”refers to the luminous fluxes being distinguishable in space. Forexample, various luminous fluxes are spatially distinguishable if theyare collimated and travel in different directions, or are focused indifferent locations. However, the invention is not intended to belimited to these precise examples, and there are many other ways inwhich luminous fluxes can be spatially distinguished from each other.

FIG. 13 is a diagram illustrating an apparatus which uses a VIPA as anangular dispersive component, instead of using diffraction gratings, toproduce chromatic dispersion. Referring now to FIG. 13, a VIPA 240 has afirst surface 242 with a reflectivity of, for example, approximately100%, and a second surface 244 with a reflectivity of, for example,approximately 98%. VIPA 240 also includes a radiation window 247.However, VIPA 240 is not limited to this specific configuration.Instead, VIPA 240 can have many different configurations as describedherein.

As illustrated in FIG. 13, a light is output from a fiber 246,collimated by a collimating lens 248 and line-focused into VIPA 240through radiation window 247 by a cylindrical lens 250. VIPA 240 thenproduces a collimated light 251 which is focused by a focusing lens 252onto a mirror 254. Mirror 254 can be a mirror portion 256 formed on asubstrate 258.

Mirror 254 reflects the light back through focusing lens 252 into VIPA240. The light then undergoes multiple reflections in VIPA 240 and isoutput from radiation window 247. The light output from radiation window247 travels through cylindrical lens 250 and collimating lens 248 and isreceived by fiber 246.

Therefore, light is output from VIPA 240 and reflected by mirror 254back into VIPA 240. The light reflected by mirror 254 travels throughthe path which is exactly opposite in direction to the path throughwhich it originally traveled. As will be seen in more detail below,different wavelength components in the light are focused onto differentpositions on mirror 254, and are reflected back to VIPA 240. As aresult, different wavelength components travel different distances, tothereby produce chromatic dispersion.

FIG. 14 is a more detailed diagram illustrating the operation of theVIPA in FIG. 13. Assume a light having various wavelength components isreceived by VIPA 240. As illustrated in FIG. 14, VIPA 240 will cause theformation of virtual images 260 of beam waist 262, where each virtualimage 260 emits light.

As illustrated in FIG. 14, focusing lens 252 focuses the differentwavelength components in a collimated light from VIPA 240 at differentpoints on mirror 254. More specifically, a longer wavelength 264 focusesat point 272, a center wavelength 266 focuses at point 270, and ashorter wavelength 268 focuses at point 274. Then, longer wavelength 264returns to a virtual image 260 which is closer to beam waist 262, ascompared to center wavelength 266. Shorter wavelength 268 returns to avirtual image 260 which is farther from beam waist 262, as compared tocenter wavelength 266. Thus, the arrangement provides for normaldispersion.

Mirror 254 is designed to reflect only light in a specific interferenceorder, and light in any other interference order should be focused outof mirror 254. More specifically, as previously described, a VIPA willoutput a collimated light. This collimated light will travel in adirection in which the path from each virtual image has a difference ofmλ, where m is an integer. The mth order of interference is defined asan output light corresponding to m.

For example, FIG. 15 is a diagram illustrating various orders ofinterference of a VIPA. Referring now to FIG. 15, a VIPA, such as VIPA240, emits collimated lights 276, 278 and 280. Each collimated light276, 278 and 280 corresponds to a different interference order.Therefore, for example, collimated light 276 is collimated lightcorresponding to an (n+2)th interference order, collimated light 278 iscollimated light corresponding to an (n+1)th interference order, andcollimated light 280 is collimated light corresponding to an nthinterference order, wherein n is an integer. Collimated light 276 isillustrated as having several wavelength components 276 a, 276 b and 276c. Similarly, collimated light 278 is illustrated as having wavelengthcomponents 278 a, 278 b and 278 c, and collimated light 280 isillustrated as having wavelength components 280 a, 280 b and 280 c.Here, wavelength components 276 a, 278 a and 280 a have the samewavelength. Wavelength components 276 b, 278 b and 280 b have the samewavelength (but different from the wavelength of wavelength components276 a, 278 a and 280 a). Wavelength components 276 c, 278 c and 280 chave the same wavelength (but different from the wavelength ofwavelength components 276 a, 278 a and 280 a, and the wavelength ofwavelength components 276 b, 278 b and 280 b). Although FIG. 15 onlyillustrates collimated light for three different interference orders,collimated lights will be emitted for many other interference orders.

Since collimated lights at the same wavelength for differentinterference orders travel in different directions and are thereforefocused at different positions, mirror 254 can be made to reflect onlylight from a single interference order back into VIPA 240. For example,as illustrated in FIG. 15, the length of a reflecting portion of mirror254 should be made relatively small, so that only light corresponding toa single interference order is reflected. More specifically, in FIG. 15,only collimated light 278 is reflected by mirror 254. In this manner,collimated lights 276 and 278 are focused out of mirror 254.

A wavelength division multiplexed light usually includes many channels.Referring again to FIG. 13, if the thickness t between first and secondsurfaces 242 and 244 of VIPA 240 is set at a specific value, thearrangement will be able to simultaneously compensate for dispersion ineach channel.

More specifically, each channel has a center wavelength. These centerwavelengths are usually spaced apart by a constant frequency spacing.The thickness t of VIPA 240 between first and second surfaces 242 and244 should be set so that all of the wavelength components correspondingto the center wavelengths have the same output angle from VIPA 240 andthus the same focusing position on mirror 254. This is possible when thethickness t is set so that, for each channel, the round-trip opticallength through VIPA 240 traveled by the wavelength componentcorresponding to the center wavelength is a multiple of the centerwavelength of each channel. This amount of thickness t will hereafter bereferred to as the “WDM matching free spectral range thickness”, or “WDMmatching FSR thickness”.

Moreover, in this case, the round-trip optical length (2nt cos θ)through VIPA 240 is equal to the wavelength corresponding to the centerwavelength in each channel multiplied by an integer for the same θ anddifferent integer, where n is the refractive index of the materialbetween first and second surfaces 242 and 244, θ indicates a propagationdirection of a luminous flux corresponding to the center wavelength ofeach channel. More specifically, as previously described, θ indicatesthe small tilt angle of the optical axis of input light (see FIG. 8).

Therefore, all of the wavelength components corresponding to the centerwavelengths will have the same output angle from VIPA 240 and thus thesame focusing position on mirror 254, if t is set so that, for thewavelength component corresponding to the center wavelength in eachchannel, 2nt cos θ is an integer multiple of the center wavelength ofeach channel for the same θ and different integer.

For example, a 2 mm physical length in round trip (which isapproximately double a 1 mm thickness of VIPA 240) and a refractiveindex of 1.5 enable all the wavelengths with a spacing of 100 GHz tosatisfy this condition. As a result, VIPA 240 can compensate fordispersion in all the channels of a wavelength division multiplexedlight at the same time.

Therefore, referring to FIG. 14, with the thickness t set to the WDMmatching FSR thickness, VIPA 240 and focusing lens 252 will cause (a)the wavelength component corresponding to the center wavelength of eachchannel to be focused at point 270 on mirror 254, (b) the wavelengthcomponent corresponding to the longer wavelength component of eachchannel to be focused at point 272 on mirror 254, and (c) the wavelengthcomponent corresponding to the shorter wavelength component of eachchannel to be focused at point 274 on mirror 254. Therefore, VIPA 240can be used to compensate for chromatic dispersion in all channels of awavelength division multiplexed light.

FIG. 16 is a graph illustrating the amount of dispersion of severalchannels of a wavelength division multiplexed light, in a case when thethickness t is set to the WDM matching FSR thickness. As illustrated inFIG. 16, all the channels are provided with the same dispersion.However, the dispersions are not continuous between the channels.Moreover, the wavelength range for each channel at which VIPA 240 willcompensate for dispersion can be set by appropriately setting the sizeof mirror 254.

If the thickness t is not set to the WDM matching FSR thickness,different channels of a wavelength division multiplexed light will befocused at different points on mirror 254. For example, if the thicknesst is one-half, one-third or some other fraction of the round tripoptical length thickness, then focusing points of two, three, four ormore channels may be focused on the same mirror, with each channel beingfocused at a different focusing point. More specifically, when thethickness t is one-half the WDM matching FSR thickness, the light fromodd channels will focus at the same points on mirror 254, and the lightfrom even channels will focus at the same points on mirror 254. However,the lights from the even channels will be focused at different pointsfrom the odd channels.

For example, FIG. 17 is a diagram illustrating different channels beingfocused at different points on mirror 254. As illustrated in FIG. 17,wavelength components of the center wavelength of even channels arefocused at one point on mirror 254, and wavelength components of thecenter wavelength of odd channels are focused at a different point. As aresult, VIPA 240 can adequately compensate for dispersion in all thechannels of a wavelength division multiplexed light at the same time.

There are several different ways to vary the value of the dispersionadded by a VIPA. For example, FIG. 18 is a diagram illustrating a sideview of an apparatus which uses a VIPA to provide variable dispersion tolight. Referring now to FIG. 18, VIPA 240 causes each differentinterference order to have a different angular dispersion. Therefore,the amount of dispersion added to an optical signal can be varied byrotating or moving VIPA 240 so that light corresponding to a differentinterference order is focused on mirror 254 and reflected back into VIPA240.

FIG. 19 is a diagram illustrating a side view of an apparatus which usesa VIPA to provide variable dispersion. Referring now to FIG. 19, therelative distance between focusing lens 252 and mirror 254 is maintainedconstant, and focusing lens 252 and mirror 254 are moved togetherrelative to VIPA 240. This movement of focusing lens 252 and mirror 254changes the shift of light returning to VIPA 240 from mirror 254, andthereby varies the dispersion.

FIGS. 20(A) and 20(B) are diagrams illustrating side views ofapparatuses which use a VIPA to provide various values of chromaticdispersion to light. FIGS. 20(A) and 20(B) are similar to FIG. 14, inthat FIGS. 20(A) and 20(B) illustrate the travel directions of a longerwavelength 264, a center wavelength 266 and a shorter wavelength 268 oflight emitted by a virtual image 260 of beam waist 262.

Referring now to FIG. 20(A), mirror 254 is a convex mirror. With aconvex mirror, the beam shift is magnified. Therefore, a large chromaticdispersion can be obtained with a short lens focal length and a smallamount of space. When mirror 254 is convex, as in FIG. 20(A), the convexshape can typically only be seen from a side view and cannot be seenwhen viewed from the top.

Referring now to FIG. 20(B), mirror 254 is a concave mirror. With aconcave mirror, the sign of the dispersion is inverted. Therefore,anomalous dispersion can be obtained with a short lens focal length anda small space. When mirror 254 is concave, as in FIG. 20(B), the concaveshape can typically only be seen from a side view and cannot be seenwhen viewed from the top.

Therefore, typically, mirror 254 would appear flat in the top view.However, it is possible for mirror 254 to also be a concave or a convexmirror when viewed by the top, thereby indicating that the mirror is a“one-dimensional” mirror.

In FIGS. 20(A) and 20(B), mirror 254 is located at or near the focalpoint of focusing lens 252.

Therefore, as described above, mirror 254 can be convex or concave inthe side view, as illustrated, for example, in FIGS. 20(A) and 20(B),respectively. A convex mirror can enhance the chromatic dispersion and aconcave mirror can reduce or even invert the chromatic dispersion fromnegative (normal) to positive (anomalous). More specifically, a convexmirror generates larger dispersion in the negative direction and aconcave mirror generates smaller dispersion in the negative direction ordispersion inverted to positive. This is possible because the magnitudeof chromatic dispersion is a function of the curvature of the mirror inthe side view.

FIG. 21 is a graph illustrating the output angle of a luminous flux fromVIPA 240 versus wavelength of the luminous flux. As can be seen fromFIG. 21, a curve 282 of the wavelength versus the output angle is notlinear.

Since the relationship between the wavelength and the output angle of aluminous flux produced by a VIPA is not linear, the chromatic dispersionis not constant in a wavelength band as long as a flat mirror, a normalconvex mirror or a normal concave mirror is used as mirror 254. Thisnonlinearity in chromatic dispersion is referred to as the higher orderdispersion.

Generally, referring to the apparatuses in FIGS. 20(A) and 20(B), thenonlinearity in chromatic dispersion can be understood by referring tothe following Equation (3):(angular dispersion)·(1−f·(1/R))∝chromatic dispersion,where f is the focal length of lens 252 and R is the radius of curvatureof mirror 254.

FIG. 22 is a graph illustrating the angular dispersion of VIPA 240versus the wavelength of a luminous flux. Generally, the curve 284 inFIG. 22 represents the slope of curve 282 in FIG. 21. As can be seenfrom FIG. 22, the angular dispersion is not constant. Instead, theangular dispersion changes as the wavelength changes.

FIG. 23 is a graph illustrating the term (1−f·(1/R)) in Equation 3,above, versus wavelength. More specifically, line 286 represents a graphof the term (1−f·(1/R)) versus wavelength for a flat mirror (radius ofcurvature equals “∞” (infinity)). Line 288 represents a graph of theterm (1−f·(1/R)) versus wavelength for a concave mirror (radius ofcurvature equals “+”). Line 290 represents a graph of the term(1−f·(1/R)) versus wavelength for convex mirror (radius of curvatureequals “−”). As illustrated in FIG. 23, each of the mirrors has aconstant radius of curvature.

FIG. 24 is a diagram illustrating the chromatic dispersion versuswavelength of an apparatus such as in FIGS. 20(A) and 20(B), when mirror254 is a convex mirror, a flat mirror and a concave mirror. Morespecifically, curve 292 is a curve of the chromatic dispersion versuswavelength when mirror 254 is a convex mirror. Curve 294 is a curve ofthe chromatic dispersion versus wavelength when mirror 254 is a flatmirror. Curve 296 is a curve of the chromatic dispersion versuswavelength when mirror 254 is a concave mirror.

In a very general manner, curves 292, 294 and 296 each represent aproduct of the angular dispersion illustrated in FIG. 22 with theappropriate line illustrated in FIG. 23, as indicated by Equation 3,above. More specifically, generally, curve 292 represents a product ofcurve 284 in FIG. 22 and line 290 in FIG. 23. Generally, curve 294represents a product of curve 284 in FIG. 22 and line 286 in FIG. 23.Generally, curve 296 represents a product of curve 284 in FIG. 22 andline 288 in FIG. 23.

As can be seen from FIG. 24, the chromatic dispersion is not constantwhether a convex, flat or concave mirror is used as mirror 254.

According to the above, this wavelength dependence of chromaticdispersion can be reduced or eliminated by chirping the curvature ofmirror 254.

More specifically, FIG. 25 is a graph illustrating a curve 298 of theterm (1−f·(1/R)) in Equation 3, above, versus wavelength. Generally,curve 298 in FIG. 25 is inverse to curve 284 in FIG. 22. Therefore, amirror having the characteristics in FIG. 25 will provide a constantchromatic dispersion, as illustrated by the curve 300 in FIG. 26.

For example, with the apparatus illustrated, for example, in FIG. 14, alonger wavelength has a larger dispersion in the negative direction thana shorter wavelength. Therefore, mirror 254 can be designed to have aconcave portion where the longer wavelength reflects, and a convexportion were the shorter wavelength reflects, to effectively cancel thewavelength dependence of dispersion. Ideally, the curvature of mirror254 varies from convex to concave continuously along the focusing pointof light when the wavelength changes from short to long. If thismodification is based on a conventional convex mirror, not a flatmirror, the curvature of the mirror can be made to vary from strongconvex to weak convex continuously along the focusing point of lightwhen the wavelength changes from short to long.

Therefore, there are many different designs for mirror 254, to provide aconstant chromatic dispersion. For example, FIG. 27 is a graphillustrating characteristics many different mirror designs. Curve 302 inFIG. 27 illustrates a mirror which continuously changes from convex toconcave as the wavelength of output light increases. Curve 304illustrates a mirror which changes from strongly convex to slightlyconvex as the wavelength of output light increases. Curve 306illustrates a mirror which changes from slightly concave to stronglyconcave as the wavelength of output light increases. Other mirrordesigns include, for example, those shown by curves 308 and 310.

There are a virtually unlimited number of mirror designs which could beused, and such designs could be graphed in FIG. 27. Moreover, mirrordesigns are not limited to those having characteristic curves with thesame slopes as those in FIG. 27.

FIGS. 28(A), 28(B), 28(C) and 28(D) illustrate the surface shape ofvarious mirrors which can be used as mirror 254. For example, FIG. 28(A)illustrates a mirror which continuously changes from convex to concave,as represented by curve 302 in FIG. 27. FIG. 28(B) illustrates a mirrorwhich continuously changes from strong convex to weak convex, asrepresented by curve 310 in FIG. 27. FIG. 28(C) illustrates a mirrorwhich continuously changes from weak concave to strong concave, asrepresented by curve 306 in FIG. 27.

Moreover, there are a virtually unlimited number of mirror designs whichcould be used. For example, FIG. 28(D) illustrates a flat mirror whichchanges to convex. FIG. 27(E) illustrates a flat mirror which changes toconcave. FIG. 28(F) illustrates a mirror having a convex portion and aconcave portion, but where the mirror does not continuously change fromconvex to concave.

Therefore, as indicated above, an apparatus includes a VIPA, a mirrorand a lens. The VIPA receives an input light and produces acorresponding output light (such as a luminous flux) propagating awayfrom the VIPA. The lens focuses the output light onto the mirror so thatthe mirror reflects the output light and the reflected light is directedby the lens back to the VIPA. The mirror has a shape which causes theapparatus to produce a constant chromatic dispersion.

For example, output light focused by the lens is incident on a differentsurface point on the mirror as the wavelength of the output lightchanges. The mirror is shaped so that the surface points changecontinuously from convex to concave as the wavelength of the outputlight changes from shorter to longer. As another example, the mirror canbe shaped so that the surface points change continuously from strongerconvex to weaker convex as the wavelength of the output light changesfrom shorter to longer.

Alternatively, the mirror can be shaped so that the surface pointschange continuously from weaker concave to stronger concave as thewavelength of the output light changes from shorter to longer. There aremany other examples. For example, the mirror can have a concave portionand a convex portion so that output light at a shorter wavelength than aspecific wavelength reflects off the convex portion and so that outputlight at a longer wavelength than the specific wavelength reflect offthe concave portion.

Moreover, for example, the mirror can have a flat portion whichcontinuously changes to a concave portion in correspondence with anincrease in the wavelength of the output light above a specificwavelength so that output light at a shorter wavelength than thespecific wavelength is incident on the flat portion and output light ata longer wavelength than the specific wavelength is incident on theconcave portion. Or, the mirror can have a convex portion whichcontinuously changes to a flat portion in corresponding with an increasein the wavelength of the output light above a specific wavelength sothat output light at a shorter wavelength than the specific wavelengthis incident on the convex portion and output light at a longerwavelength than the specific wavelength is incident on the flat portion.

A VIPA, as described above, provides a much larger angular dispersionthan a diffraction grating. Therefore, a VIPA can be used to compensatefor much larger chromatic dispersion than a spatial grating pairarrangement as illustrated in FIGS. 6(A) and 6(B).

A mirror, as described above, to reflect light back to a VIPA tocompensate for chromatic dispersion, can be described as a cylindricalmirror since the mirror shape is that of the surface of a cylinder. Inother words, as shown in FIG. 29, the mirror has the same radius ofcurvature along an axis forming the cylinder. Since chromatic dispersionis a function of the radius of the mirror curvature as described above,the chromatic dispersion will not change when the mirror is moved alongthe axis forming the cylinder. As shown in FIG. 30(A), the chromaticdispersion may change within each channel as previously described (seeFIG. 24). However, the chromatic dispersion will be periodic, as shownin FIG. 30(B), and the chromatic dispersion will be approximately thesame for all the channels.

FIG. 31(A) is a graph illustrating chromatic dispersion versuswavelength for one channel of a wavelength division multiplexed light,after undergoing chromatic dispersion compensation with a VIPA with amodified cylindrical mirror as in, for example, FIGS. 28(A) through28(F). Referring now to FIG. 31(A), it can be seen that the amount ofchromatic dispersion is substantially the same for each wavelengthwithin the same channel.

FIG. 31(B) is a graph illustrating chromatic dispersion versuswavelength for all wavelengths (and therefore, many channels) of awavelength division multiplexed light, after undergoing chromaticdispersion compensation with a VIPA with a modified cylindrical mirroras in, for example, FIGS. 28(A) through 28(F). Referring now to FIG.31(B), it can be seen that the amount of chromatic dispersion issubstantially the same, or uniform, for all wavelengths in all channels.

FIG. 32 is a diagram illustrating a top view of an apparatus using aVIPA to provide variable chromatic dispersion to light, according to afurther embodiment of the present invention. Referring now to FIG. 32, acone shaped mirror 400 is used to reflect light back to VIPA 240. Mirror400 is movable in a direction 401.

As previously indicated, VIPA 240 produces a collimated luminous flux,which can be referred to as a collimated output light, traveling in adirection determined by the wavelength of the light. The angulardispersion direction of VIPA 240 is the direction in which the travelingdirection of the collimated output light changes as the wavelength ofthe light changes, and is represented, for example, by direction 402 inFIG. 32. Collimated output lights for different wavelengths will be inthe same plane.

Therefore, direction 401 is along the surface of the cone and can bedescribed as being perpendicular to both the angular dispersiondirection of VIPA 240 and the traveling directions of collimated lightfrom VIPA 240. Alternatively, direction 401 can be described as beingperpendicular to a plane which includes the traveling direction of thecollimated output lights for different wavelengths from VIPA 240.

FIGS. 33(A) and 33(B) are diagrams illustrating how mirror 400 can beformed, for example, from a section of a cone 405, according to anembodiment of the present invention. As can be seen from FIG. 33(A),direction 401 preferably passes along the surface of, and through thetop of, cone 405. Although it is preferable for direction 401 to passthrough the top of cone 405, it is not necessary to pass through thetop.

In FIG. 33(B), mirror 400 is shown with three different radii ofcurvature A, B and C. Radius of curvature A is the largest, radius ofcurvature C is the smallest, and radius of curvature B is between A andC in size.

By moving the mirror in direction 401 (corresponding, for example, todirection 401 in FIG. 32), the position of the light focus moves from Ato C on the surface of the cone shaped mirror in FIG. 33(B). Since theradii are different for A, B and C, the chromatic dispersion will bedifferent. Thus, the chromatic dispersion will be varied by moving thecone shaped mirror.

FIG. 34(A) is a graph illustrating the amount of chromatic dispersionversus wavelength within one channel for radii of curvature A, B and Cof a cone shaped mirror when the mirror is moved in a direction such asdirection 401, according to an embodiment of the present invention. Ascan be seen from FIG. 34(A), generally, radius of curvature C producesthe greatest amount of chromatic dispersion. Generally, radius ofcurvature A produces the smallest amount of chromatic dispersion. As canbe seen from FIG. 34(A), the amount of chromatic dispersion produced byradius of curvature B is between A and C.

As can be seen from FIG. 34(A) and also described with reference toFIGS. 24 and 30(A), the amount of chromatic dispersion will be differentfor different wavelengths within a channel. However, as described withreference to FIGS. 26, 31(A) and 31(B), by modifying the mirror, it ispossible to provide a uniform amount of chromatic dispersion in eachchannel, and in all the channels.

For example, FIG. 34(B) is a diagram illustrating the radii of curvatureA, B and C when a cone shaped mirror is moved in a direction such asdirection 401, according to an embodiment of the present invention. Bycontrast, FIG. 34(C) is a diagram illustrating modified radii ofcurvature A′, B′ and C′ when a modified cone shaped mirror to provideuniform chromatic dispersion is moved in a direction such as direction401, according to an embodiment of the present invention. For example,in the modified mirror, output light focused by lens 252 is incident ona different surface point on the mirror as the wavelength of the outputlight changes. The mirror is shaped so that the surface points changecontinuously from convex to concave as the wavelength of the outputlight changes from shorter to longer. As another example, the mirror canbe shaped so that the surface points change continuously from strongerconvex to weaker convex as the wavelength of the output light changesfrom shorter to longer.

Alternatively, the mirror can be shaped so that the surface pointschange continuously from weaker concave to stronger concave as thewavelength of the output light changes from shorter to longer. There aremany other examples. For example, the mirror can have a concave portionand a convex portion so that output light at a shorter wavelength than aspecific wavelength reflects off the convex portion and so that outputlight at a longer wavelength than the specific wavelength reflect offthe concave portion.

As a result, the modified mirror will provide a uniform chromaticdispersion in each channel, and in all the channels.

FIG. 35 is a graph illustrating the chromatic dispersion versuswavelength in one channel for radii of curvature A′, B′ and C′,according to an embodiment of the present invention. As can be seen fromFIG. 35, each radii of curvature A′, B′ and C′ produces a uniform butdifferent amount of chromatic dispersion. Therefore, each channel willhave a uniform chromatic dispersion and the amount of the chromaticdispersion is variable by moving the mirror.

FIG. 36 is a diagram illustrating various angles in an apparatus whichuses a VIPA, according to an embodiment of the present invention.Referring now to FIG. 36, Θ and θ are the average incident angles, and Φand φ are the output angles with respect to the normal line to a plate,such as second surface 244, forming VIPA 240. Θ and Φ indicate theangles in air, whereas θ and φ indicate the angles in glass betweensurfaces 242 and 244 of VIPA 240. The angles in the air areapproximately n times larger than those in the glass because of therefraction at the glass surface. Here, n is the index of the glass.

FIG. 37 is an additional diagram illustrating angles in an apparatuswhich uses a VIPA, according to an embodiment of the present invention.As indicated in FIG. 37, the output angle φ is determined as thedirection where the difference in the light paths originating at twoadjacent beam waists is a multiple of the light wavelength. The spacingbetween the adjacent beam waists is 2t (t is the thickness of the VIPA,as illustrated, for example, in FIG. 8) and the output angle in theglass is φ. So, 2t cos φ=mλ/n (m is an integer). From this, the angulardispersion is dΦ/dλ=−n²/λΦ, as shown by the following Equation (4):

$\begin{matrix}{{{Separation}\mspace{14mu}{of}\mspace{14mu}{the}\mspace{14mu}{light}\mspace{14mu}{paths}\text{:}\mspace{14mu} d} = {2t\;\sin\mspace{11mu}\phi}} & \; \\{{{Difference}\mspace{14mu}{of}\mspace{14mu}{the}\mspace{14mu}{path}\mspace{14mu}{lengths}\text{:}\mspace{14mu}{m \cdot \frac{\lambda}{n}}} = {2t\;\cos\mspace{11mu}\phi}} & \; \\{{\frac{m\;\Delta\;\lambda}{n} = {{- 2}t\;\sin\mspace{11mu}\phi\;\Delta\;\phi}}} & \; \\{{\Delta\;\phi} = {{{- \cot}\mspace{11mu}{\phi \cdot \frac{\Delta\;\lambda}{\lambda}}} \approx {{- \frac{1}{\phi}}\frac{\Delta\;\lambda}{\lambda}}}} & \; \\\begin{matrix}{\theta\text{:}\mspace{14mu}{Input}\mspace{14mu}{angle}\mspace{14mu}{in}\mspace{14mu}{glass}} & \; & \; & {\phi\text{:}\mspace{14mu}{Output}\mspace{14mu}{angle}\mspace{14mu}{in}\mspace{14mu}{glass}}\end{matrix} & \; \\\begin{matrix}{\Theta\text{:}\mspace{14mu}{Input}\mspace{14mu}{angle}\mspace{14mu}{in}\mspace{14mu}{air}} & \; & \; & {\mspace{20mu}{\Phi\text{:}\mspace{14mu}{Output}\mspace{14mu}{angle}\mspace{14mu}{in}\mspace{14mu}{air}}}\end{matrix} & \; \\{\Theta \approx {n\;\theta}} & \; \\{\Phi \approx {n\;\phi}} & \; \\{{\Delta\;\Phi} \approx {\Delta\;\phi}} & \; \\\begin{matrix}{{{\Delta\;\Phi} \approx {{- \frac{n}{\phi}}\frac{\Delta\;\lambda}{\lambda}} \approx {{- \frac{n^{2}}{\Phi}}\frac{\Delta\;\lambda}{\lambda}}},} & \; & {\frac{\mathbb{d}\Phi}{\mathbb{d}\lambda} \approx {- \frac{n^{2}}{\lambda\;\Phi}}}\end{matrix} & {{Equation}\mspace{14mu}(4)}\end{matrix}$

FIG. 38 is a diagram illustrating how chromatic dispersion is generatedin an apparatus using a VIPA, according to an embodiment of the presentinvention. FIG. 14 also illustrates how chromatic dispersion isgenerated, but FIG. 38 is a more quantitative description.

Referring now to FIG. 38, the light travel angle in the air with respectto the normal line to the VIPA is Φ−Θ. Also, the focal length of lens252 is f and the depth of the center beam waist is a. The light focusingposition y on the mirror is y=f(Φ−Θ). The mirror shape is c(y) as afunction of y. The mirror slope h is dc/dy. Then, the beam shift afterthe round trip is obtained by the following Equation (5):

$\begin{matrix}{{{Mirror}\mspace{14mu}{shape}\text{:}\mspace{14mu}{c(y)}},} & \; \\{{{{Slope}\mspace{14mu}{of}\mspace{14mu}{mirror}\mspace{14mu}{surface}\text{:}\mspace{14mu}{h(y)}} = \frac{\mathbb{d}{c(y)}}{\mathbb{d}y}},{y \approx {f\left( {\Phi - \Theta} \right)}}} & \; \\{\left( {{Beam}\mspace{14mu}{shift}} \right) \approx {{2\left( {f - a} \right)\left( {\Phi - \Theta} \right)} + {2f\;{h(y)}}}} & \; \\{({Delay}) = {{\frac{n}{c}\left( {{Distance}\mspace{14mu}{change}} \right)} \approx {\frac{n}{c} \cdot \frac{\left( {{Beam}\mspace{14mu}{shift}} \right)}{\phi}}}} & {{Equation}\mspace{14mu}(5)} \\{\mspace{79mu}{\approx {\frac{2n^{2}}{c\;\Phi}\left\{ {{\left( {f - a} \right)\left( {\Phi - \Theta} \right)} + {f\;{h(y)}}} \right\}}}} & \;\end{matrix}$

The distance change in FIG. 38 is easily obtained from the beam shift,and the delay is the distance change divided by the speed of light inthe glass. The chromatic dispersion is calculated as the delay changewith the wavelength change and is shown by the following Equation (6):

$\begin{matrix}{({Dispersion}) = \frac{\mathbb{d}({Delay})}{\mathbb{d}\lambda}} & {{Equation}\mspace{14mu}(6)} \\{\mspace{130mu}{\approx {\frac{2n^{2}}{c}\left\{ {{\left( {f - a} \right)\frac{\Theta}{\Phi^{2}}} + {\frac{f}{\Phi}\frac{\mathbb{d}{h(y)}}{\mathbb{d}y}\frac{\mathbb{d}y}{\mathbb{d}\Phi}} - \frac{f\;{h(y)}}{\Phi^{2}}} \right\}\frac{\mathbb{d}\Phi}{\mathbb{d}\lambda}}}} & \; \\{\mspace{130mu}{\approx {{- \frac{2n^{4}}{c\;\lambda\;\Phi^{3}}}\left\{ {{\left( {f - a} \right)\;\Theta} + {f^{2}\Phi\;\frac{\mathbb{d}{h(y)}}{\mathbb{d}y}} - {f\;{h(y)}}} \right\}}}} & \; \\{{For}\mspace{14mu} a\mspace{14mu}{cylindrical}\mspace{14mu}{mirror}\mspace{14mu}{of}\mspace{14mu}{radius}\mspace{14mu} r\text{:}} & \; \\{({Dispersion}) \approx {{- \frac{2n^{4}\Theta}{c\;\lambda\mspace{11mu}\Phi^{3}}}\left\{ {f - a + \frac{f^{2}}{r}} \right\}}} & {{Equation}\mspace{14mu}(7)}\end{matrix}$

If the mirror is a cylindrical mirror and has a circular shape alongwith angular dispersion direction, dh/dy is simply 1/r and the followingEquation (7) is obtained:

From Equation (7), it can be seen that chromatic dispersion is notuniform in a WDM channel and, instead, the chromatic dispersion changesapproximately in proportion to 1/Φ³.

As indicated in Equation (6), chromatic dispersion is a function of Φ.To make this dispersion uniform in a WDM channel, this formula needs tobe constant as Φ changes. Therefore, the value in the large parenthesisof Equation (6) should be proportional to Φ³ (small change of λ isignored). Assuming the proportional constant is K (this means thechromatic dispersion is −2n⁴K/cλ) and that n, c, λ, f and a are constantor almost constant for the small change of wavelength, we get thefollowing Equation (8).

$\begin{matrix}{{{\left( {f - a} \right)\Theta} + {f^{2}\Phi\frac{\mathbb{d}{h(y)}}{\mathbb{d}y}} - {{fh}(y)}} = {K\;{\Phi^{3}.}}} & {{Equation}\mspace{14mu}(8)} \\{{Here},{y \approx {{f\left( {\Phi - \Theta} \right)}.\mspace{14mu}{So}}},{\Phi = {\frac{y}{f} + \Theta}}} & \;\end{matrix}$The condition for a uniform dispersion in a WDM channel is

${{{f^{2}\left( {\frac{y}{f} + \Theta} \right)}\frac{\mathbb{d}{h(y)}}{\mathbb{d}y}} - {{fh}(y)}} = {{k\left( {\frac{y}{f} + \Theta} \right)}^{3} - {\left( {f - a} \right)\Theta}}$The mirror slope h should be zero at the center y=0. Equation (8) issolved to get the following Equation (9):

$\begin{matrix}{{h(y)} = {{\frac{K}{2f^{4}}y^{3}} + {\frac{3K\;\Theta}{2f^{3}}y^{2}} + {\frac{{K\;\Theta^{2}} - \left( {f - a} \right)}{f^{2}}y}}} & {{Equation}\mspace{14mu}(9)}\end{matrix}$

The mirror curve is obtained after integration and is shown by thefollowing Equation (10):

$\begin{matrix}\begin{matrix}{{c(y)} = {\int{{h(y)}{\mathbb{d}y}}}} \\{= {{\frac{K}{8f^{4}}y^{4}} + {\frac{K\;\Theta}{2f^{3}}y^{3}} + {\frac{{K\;\Theta^{2}} - \left( {f - a} \right)}{2f^{2}}y^{2}}}}\end{matrix} & {{Equation}\mspace{14mu}(10)}\end{matrix}$Equation (10) determines the ideal curves for different K, which weredescribed, for example, in FIG. 28.

The mirror shape is determined by the value K, which gives the chromaticdispersion. To get the shape along the curve A, B, and C in FIG. 33(B),a small K, a medium K, and a large K can be used, respectively, forEquation (10). The curves are illustrated in FIGS. 39(A), 39(B) and39(C). However, for easy manufacturing, the shapes could beapproximately a part of an ellipse, or a parabola, or a hyperbola. Inthese cases, the mirror can be made as a part of a cone.

FIG. 40 is a diagram illustrating an example of a cone for forming amirror, according to an embodiment of the present invention. Referringnow to FIG. 40, cone 405 has a base 406. If base 406 is a circle, cone405 is a normal cone. However, cone 405 may be stretched, for example,in a side direction. In such case, base 406 will be ellipse, as shown inFIG. 40. In the case of ellipse, base 406 has a longer axis r₁ and ashorter axis r₂. Direction 401 is determined by the line passing alongthe cone surface from the top of the cone to the bottom where the conesurface hits the longer or shorter axis in base 406. However, this linedoes not necessarily have to hit one of the axes. As shown in FIG. 40,cone 406 is cut by a plane 407 which is perpendicular to direction 401.A cut curve 408 for the mirror is an ellipse, parabola or hyperbola,depending on the top angle of cone 405. Therefore, cut curve 408 in themirror area is a part of one of these three curves. A modified coneshaped mirror is defined so that cut curve 408 is determined by Equation(10), rather than the three shapes.

Light for different WDM channels will be focused at different positionsdisplaced in the direction 401. Therefore, the different WDM channelswill see different curves and generate different chromatic dispersion.Therefore, the cone shape can be further modified so that the cut curvesfor different WDM channels are determined by Equation (10) withdesirable value Ks. This indicates that the dispersion change is notlimited to a linear change with wavelength or WDM channels and it couldchange in any way.

FIG. 41 is a diagram illustrating a step shaped mirror surface,according to an embodiment of the present invention. This mirror canprovide different shapes for different WDM channels without causing anexcess tilt of the mirror with respect to the incident light.

Referring again to FIG. 32, mirror 400 is movable in direction 401.Mirror 400 can also be described as movable in or around a focal planeof lens 252. Mirror 400 has a cone shape, or modified cone shape, asdescribed above, so mirror 400 will have different curvatures along thesurface. Since the curvature changes along direction 401, and mirror 400is movable in this direction, the chromatic dispersion can be varied bymoving mirror 400 by a relatively small distance. In this design, themoving distance of mirror 400 would typically be less than 1 cm, whichis much smaller than the moving distance of mirror 254 in FIG. 19.

Further, in FIG. 19, the position of lens 252 is movable, whereas inFIG. 32, the position of lens 252 would typically be fixed. Therefore,in FIG. 19, a large space will be required between VIPA 240 and lens252, so that the lens 252 and mirror 254 can be moved together for arelatively large distance to provide the required amount of chromaticdispersion. This large space between VIPA 240 and lens 252 isundesirable, and greatly increases the overall size of the apparatus. Bycomparison, in FIG. 32, a relatively small space is required betweenVIPA 240 and lens 252, and mirror 400 only has to move a relativelysmall distance to provide the required amount of chromatic dispersion,thereby allowing the overall apparatus to be much smaller than that inFIG. 19.

FIG. 42 is a diagram illustrating a side view of an apparatus using aVIPA to provide chromatic dispersion slope, according to an additionalembodiment of the present invention. Referring now to FIG. 42, anangular dispersive component 500 is positioned between VIPA 240 and lens252. Angular dispersive component 500 could be, for example, atransmission type diffraction grating, a reflection type diffractiongrating or a holographic grating.

Angular dispersive component 500 has an angular dispersion directionwhich is perpendicular to the angular dispersion direction of VIPA 240.

Preferably, the amount of angular dispersion provided by angulardispersive component 500 should be large enough to distinguish thedifferent wavelengths for different WDM channels. Therefore, preferably,the angular dispersion provided by angular dispersive component 500should be larger than approximately 0.1 degrees/nm. This number isreadily achievable by using a diffraction grating as angular dispersivecomponent 500. However, the present invention is not limited to anyparticular amount of angular dispersion.

In FIG. 42, the position of mirror 400 is preferably fixed. This isdifferent than in FIG. 32, where the position of mirror 400 is movable.However, in FIG. 42, mirror 400 is not limited to being fixed, and canbe movable to add variable dispersion.

By using angular dispersive component 500 between VIPA 240 and lens 252,the light in different channels will be focused by lens 252 at positionswhich are displaced along direction 401 (not shown in FIG. 42) on thesurface of mirror 400 because of the angular dispersion of angulardispersive component 500, and will see a different curvature of mirror400. As a result, different channels will have different chromaticdispersions. This channel dependent chromatic dispersion is called highorder dispersion or dispersion slope, and is required for compensationof a fiber dispersion since different WDM channels traveling in a fiberwill see different chromatic dispersion in the fiber.

FIG. 43(A) is a graph illustrating the amount of chromatic dispersionfor all wavelengths (many channels) with a cone shaped mirror used asmirror 400 in FIG. 42, according to an embodiment of the presentinvention. For example, this cone shaped mirror would typically be asillustrated in FIGS. 33(A) and 33(B). As illustrated in FIG. 43(A), theamount of chromatic dispersion is not uniform in each channel anddiffers for different channels.

FIG. 43(B) is a graph illustrating the amount of chromatic dispersionfor all wavelengths (many channels) with a modified cone shaped mirrorused as mirror 400 in FIG. 42, according to an embodiment of the presentinvention. For example, this modified cone shaped mirror would typicallyhave radii of curvature A′, B′ and C′ as in FIG. 34(C), according to anembodiment of the present invention. As illustrated in FIG. 43(B), theamount of chromatic dispersion is uniform in each channel and differentfor different channels.

In FIGS. 43(A) and 43(B), the dispersion is shown as increasing withincreasing wavelength. However, in some embodiments of the presentinvention, the dispersion could decrease with increasing wavelength byinverting angular dispersive component 500 or by inverting the directionof the cone shaped mirror.

Therefore, parameters (such as the mirror shape, lens focal length,etc.) are preferably designed so that the chromatic dispersion for eachWDM channel, such as those shown, for example, in FIG. 43(A) or 43(B),is the same amount but opposite sign to the chromatic dispersion of thetransmission line at the corresponding wavelength for the purpose of thesimultaneous dispersion compensation of all WDM channels. Namely,although different WDM channels may experience different chromaticdispersion amounts through the transmission line, a VIPA can be used, asdescribed herein, to compensate for the dispersion of the WDM channelswith different dispersion amounts.

FIG. 44 is a diagram illustrating the use of a holographic grating 510as an angular dispersive component between VIPA 240 and lens 252,according to an embodiment of the present invention.

Moreover, FIG. 45 is a diagram illustrating the use of a reflection typegrating 520 as an angular dispersive component between VIPA 240 and lens252, according to an embodiment of the present invention.

When a diffraction grating is used as an angular dispersive component(see FIG. 42), one problem is its polarization dependence. Therefore, aquarter wave plate can be used to cancel the polarization dependence ofthe diffraction grating.

For example, FIG. 46 is a diagram illustrating the use of a quarter waveplate 530 inserted between the diffraction grating and lens 252.

FIG. 47 is a diagram illustrating the use of quarter wave plate 530inserted between lens 252 and the cone shape mirror 400. As an example,quarter wave plate 530 is positioned with the axes at 45° with respectto the plane of s or p polarization of the diffraction grating.

With configurations as in FIGS. 46 and 47, light passed through thediffraction grating with p-polarization will return to the diffractiongrating with s-polarization, and light passed through the diffractiongrating with s-polarization will return to the diffraction grating withp-polarization. Therefore, the polarization dependence of thediffraction grating is canceled.

FIG. 48(A) is a diagram illustrating a side or top view of an apparatuswhich uses a VIPA to provide two different chromatic dispersions fordifferent channels, according to a still further embodiment of thepresent invention. Referring now to FIG. 48(A), a wavelength filter 510is positioned between lens 252 and mirrors M1 and M2. Wavelength filter510 filters the light from lens 252 so that light at wavelength λ1 isdirected to mirror M1, and light at wavelength λ2 is directed to mirrorM2. Mirror M1 has a different curvature than mirror M2 and therefore, λ1and λ2 will have different chromatic dispersion. Thus, each of mirrorsM1 and M2 can be, for example, a cylindrical mirror or a modifiedcylindrical mirror, as described herein. For example, mirrors M1 and M2can be modified cylindrical mirrors to provide uniform but differentamount of chromatic dispersion in channels corresponding to λ1 and λ2.

FIG. 48(B) is a graph illustrating chromatic dispersion versuswavelength for the apparatus in FIG. 48(A), where mirrors M1 and M2 aremodified cylindrical mirrors to provide uniform chromatic dispersionwithin each channel, according to an embodiment of the presentinvention.

While FIG. 48(A) shows an apparatus configured for two wavelengths,there is generally no limit in the number of wavelength filters andmirrors which can be used to separate additional wavelengths orchannels.

For example, FIG. 49 is a diagram illustrating a side or top view of anapparatus which uses a VIPA to provide three different chromaticdispersion for different channels, according to an embodiment of thepresent invention. Referring now to FIG. 49, wavelength filters 520 and530 are used to direct light at wavelengths λ1, λ2 and λ3 to mirrors M1,M2 and M3, respectively.

According to the above embodiments of the present invention, anapparatus which uses a VIPA in combination with a mirror, such as a coneor modified cone shaped mirror, to generate dispersion slope or higherorder dispersion. The cone or modified cone shape of the mirror isdesigned so that the dispersion slope or higher order dispersion of theapparatus compensates for dispersion slope or higher order dispersion ofa transmission line (fiber).

In an optical communication system in which a transmitter transmits anoptical signal through a transmission line to a receiver, the apparatusof the present invention can be inserted in the transmitter, thetransmission line, the receiver, or in any combination of thetransmitter, transmission line and receiver. For example, in FIG. 1, theapparatus of the present invention can be inserted in transmitter 30,optical fiber 34 (for example, a transmission line) or receiver 36, orin any combination of transmitter 30, optical fiber 34 and receiver 36.Further, two or more of the apparatuses of the present invention can becascaded together, or only one apparatus can be used in transmitter 30,optical fiber 34 and/or receiver 36. Thus, the present invention is notlimited to the number of apparatuses which can be used together toprovide the required affect.

One problem with an apparatus which use a VIPA to provide chromaticdispersion, as in the above-described embodiments of the presentinvention, is that the apparatus has a relatively narrow band in thetransmission spectrum. Generally, the band is narrow due to insertionloss from fiber-to-fiber. For example, in FIG. 13, insertion loss occursfrom the light traveling out of fiber 246 to when the light is againreceived by fiber 246 after traveling through VIPA 240 and beingreflected by mirror 254.

For example, FIG. 50 is a graph illustrating the insertion loss in anapparatus which uses a VIPA to provide chromatic dispersion, accordingto an embodiment of the present invention. Referring now to FIG. 50,curve 550 illustrates the actual insertion loss which might typicallyoccur for one channel. By contrast, curve 560 illustrates a moredesirable insertion loss for the channel.

The insertion loss is due to several different factors, one major factoris a loss due to different diffraction efficiency at differentwavelengths.

For example, FIG. 51 is a diagram illustrating different diffractionefficiency at different wavelengths. Referring now to FIG. 51, lightoutput from VIPA 240 is focused by lens 252 on a mirror 570. Light atthe shortest wavelength is focused at point 580, light at the centerwavelength is focused at point 590, and light at the longest wavelengthis focused at point 600. However, due to the characteristics of VIPA240, and especially to the physics underlying the multiple reflectionincurring inside VIPA 240, the light at the center wavelength at point590 will be the strongest, whereas the light at the shortest wavelengthand the longest wavelength at points 580 and 600, respectively, will beweaker.

For example, FIG. 52 is a diagram illustrating the light intensity oflight traveling out of a fiber and into a VIPA in the above embodimentsof the present invention. FIG. 52 includes fiber 246 and lenses 248 and250 as in FIG. 13, but the VIPA is removed and the light is allowed totravel to a screen 610. A dotted box 240 shows where the VIPA would bepositioned.

As indicated in FIG. 52, the light has a light intensity shown by curve620 at screen 610. As a result, the insertion loss can be made closer tothe desired insertion loss 560 in FIG. 50 if the far field distributionof the input light provided to the VIPA is a double-humped shape. Inthis manner, the transmission spectrum of the apparatus will be muchflatter.

FIG. 53 is a diagram illustrating a side view of an optical phase maskon an input fiber to produce a double-humped shape far fielddistribution, in an apparatus which uses a VIPA to provide chromaticdispersion, according to an embodiment of the present invention.Referring now to FIG. 53, an input fiber 246 (corresponding, forexample, to input fiber 246 in FIG. 13) has a core 650. Optical phasemasks 660 and 670 cover a portion of the top and bottom, respectively,of core 650. As a result, a double-humped shape far field distributionwill be provided at the input to the VIPA (not illustrated in FIG. 53),and the insertion loss of the apparatus will have a more desirableinsertion loss.

FIG. 54 is a diagram illustrating a cross-sectional view along lines54—54 in FIG. 53, according to an embodiment of the present invention.As can be seen from FIGS. 53 and 54, phase masks 660 and 670 cover thetop and bottom, respectively. The phase masks should not be on the sideportions of the core.

It is not necessary for the phase masks to be on the input fiber.Instead, for example, the phase masks could be on the VIPA.

For example, FIG. 55 is a diagram illustrating a side view of phasemasks on a VIPA to provide a double-humped shape far field distributionwith respect to light received inside the VIPA, according to anembodiment of the present invention. Elements in FIG. 55 are similar tothat in FIG. 11.

Referring now to FIG. 55, optical phase masks 690 and 695 are positionedon the light incident window surface 124, to provide a double-humpedshape far field distribution of light received into the VIPA.

FIG. 56 is a diagram illustrating a side view of phase masks on a VIPAto provide a double-humped shape far field distribution with respect tolight received inside the VIPA, according to an additional embodiment ofthe present invention. FIG. 56 is different than FIG. 55 in that phasemasks 690 and 695 are provided on reflecting surface 122. Therefore,phase masks can be on either reflecting surface or on the light incidentwindow of the VIPA.

Further, a double-humped shape far field distribution can be obtained bypositioning phase masks in the center of the input light.

For example, FIGS. 57 and 58 are diagrams illustrating a side view ofphase masks on a VIPA to provide a double-humped shape far fielddistribution with respect to light received inside the VIPA, accordingto an additional embodiment of the present invention. In FIGS. 57 and58, a phase mask 700 is positioned in the center of the input light. Inthis case, the optical phase at the center of the far field distributionmay be π, and may be 0 at the ends. This is the opposite of the farfield distribution in FIGS. 53–56.

As indicated above, phase masks can be used to provide a double-humpedshape far field distribution. The phase mask preferably has a thicknesscorresponding to the addition of π to the optical phase. However, apreferable range of optical phase added by the phase mask is ⅔π to 4/3π.

Any transparent material that provides the proper additional phase canbe used for the phase mask. For example, SiO₂ would be a typicalmaterial for use as a phase mask.

As indicated above, a phase mask is used to provide a double-hump shapedfar field distribution. Here, a “double-humped shape” is defined ashaving two almost identical peaks with a valley between the peaks. Thedepth of the valley should be less than or equal to 50% of the top peakvalue, and preferably less than 20% of the top peak value. Preferably,the peaks are identical, but it is satisfactory for the peaks to have anamplitude of within 10% of each other.

Further, instead of using a phase mask, there are other ways to producea double-hump shaped far field distribution, and the present inventionis not limited to using a phase mask for this purpose.

The above described embodiments using a phase mask to produce adouble-hump shaped far field distribution are applicable to embodimentsof the present invention that use a VIPA to produce chromaticdispersion. However, these embodiments are also applicable to the use ofa VIPA as a demultiplexer. For example, the above-described embodimentsof the present invention relating to the use of a phase mask to producea double-hump shaped far field distribution can be applied to the VIPAin FIGS. 7 and 8.

As described above, an apparatus using a VIPA to compensate forchromatic dispersion would typically have a loss curve in each WDMchannel as shown in FIG. 50. As described above, this loss curve can beflattened by using an optical phase mask. However, there are other waysto flatten the loss curve, such as by adding excess loss.

For example, FIG. 59 is a diagram illustrating excessive loss added tothe loss curve, according to an embodiment of the present invention.Referring now to FIG. 59, by adding excess loss 705, loss curve 550 willbe flattened to become curve 710.

FIG. 60 is a diagram illustrating the use of an excess loss component toprovide excess loss, and thereby flatten the loss curve, according to anembodiment of the present invention. Referring now to FIG. 60, a VIPAdispersion compensator 720 represents an apparatus which uses a VIPA toproduce chromatic dispersion, as described herein. An excess losscomponent 730 is cascaded with VIPA dispersion component 720. Excessloss component 730 could be either upstream or downstream of VIPAdispersion component 720 and there might be some optical componentsbetween VIPA dispersion component 720 and excess loss component 730.Thus, the present invention is not limited to any specific placement ofVIPA dispersion component 720 with respect to excess loss component 730.

Excess loss component 730 can be, for example, an optical interferometeror a wavelength filter. However, a Mach-Zehnder interferometer or aFabry-Perot interferometer would be suitable, because they have aperiodic transmission curve and the period can be adjusted to the WDMchannel spacing by choosing appropriate parameters of theinterferometer. Therefore, the overall transmission curve will beflattened for all the WDM channels simultaneously.

The above described embodiments using an excess loss component areapplicable to embodiments of the present invention that use a VIPA toproduce chromatic dispersion. However, these embodiments are alsoapplicable to the use of a VIPA as a demultiplexer. For example, theabove-described embodiments of the present invention relating to the useof an excess loss component can be applied with the VIPA in FIGS. 7 and8.

Instead of using an excessive loss component, there are other ways toflatten the loss curve.

For example, FIG. 61 is a diagram illustrating a side view of a mirrorfor use with a VIPA to provide chromatic dispersion, and which willflatten the loss curve, according to an embodiment of the presentinvention. Referring now to FIG. 61, a mirror 704 could be a cone shapedmirror, a modified cone shaped mirror, a flat mirror, or any other shapemirror. FIG. 61 shows positions P, Q and R in the side view. PositionsP, Q and R corresponds, respectively to points 274, 270 and 272,respectively, in FIG. 14. Light at a shorter wavelength is focused atpoint 274 or P, and light at a longer wavelength is focused at point 272or R.

The reflectivity on mirror 740 is modulated along the angular dispersiondirection of the VIPA. That is, the reflectivity at the position Q islowest, to thereby provide a higher loss, and the reflectivity at theposition P and R is higher, to thereby provide a lower loss. Therefore,the power of the reflected light is reduced near the center of the WDMchannel, and thus the loss curve is flattened. To modify thereflectivity, a layer of light absorbing material may be coated nearposition Q or, in the case of a multi-layer mirror, the thickness of oneor more layer may be modulated.

This modulation of the reflectivity can be effectively achieved bypatterning the mirror instead of actually modulating the reflectivity,if the VIPA is used with a mirror which is not a cone or modified coneshape, that is, if the VIPA is used with a mirror such as, for example,mirror 254 in FIGS. 14, 20(A), 20(3), or the mirror shapes in FIGS.28(A) through 28(F).

For example, FIG. 62 is a diagram illustrating a front view of a mirror750, according to an embodiment of the present invention. Referring nowto FIG. 62, mirror 750 is patterned as illustrated in the figure, tochange the reflectivity of mirror 750. Here, the width of mirror 750 issmaller than the focused beam size 760 near the position Q, andtherefore, the light power reflected from near position Q is reduced.

FIGS. 63(A), 63(B) and 63(C) are diagrams illustrating another way tomodulate the effective reflectivity in the case of a VIPA used with amirror 770 which is not a cone or modified cone shape, according to anembodiment of the present invention. More specifically, FIGS. 63(A),63(B) and 63(C) illustrate a top view of incident beam 780 on mirror 770at positions P, Q and R, respectively. As illustrated in FIGS. 63(A),63(B) and 63(C), instead of modulating the reflectivity, the mirrorangle in the top view is changed. In previously described embodiments ofthe present invention, such as that in FIG. 14, the mirror is preferablyperpendicular to the average light incident angle in the top view.However, if the mirror is tilted in the top view, as in FIGS. 63 (A),63(B) and 63(C), the reflected light is deflected and the couplingefficiency to the output fiber is reduced. At positions P and R,incident light 780 is perpendicular to mirror 770 and the light is fullyreturned to the output fiber. On the other hand, at position Q, mirror770 is tilted in the top view and the reflected light is slightlydiverged from the output fiber direction. This causes an excess loss andflattening of the loss curve. By changing this tilting angle of mirror770 in the top view gradually along the angular dispersion direction ofthe VIPA, the excess loss to flatten the loss curve can be effectivelyproduced.

The changing of the mirror angle as in FIGS. 63(A), 63(B) and 63(C), andthe patterning of the mirror as in FIG.62, could be used in theabove-described apparatuses which use a VIPA in combination with amirror which is not cone or modified cone shaped. This is because, inthe case of a cone or modified cone shaped mirror, the light at awavelength may be focused effectively at different positions on themirror in the top view, and therefore, the mirror should not bepatterned or tilted in the top view.

FIG. 64 is a diagram illustrating the use of a grating between a VIPAand a lens, according to an embodiment of the present invention. FIG. 64is similar to FIGS. 44 and 45. However, whereas the embodiments in FIGS.44 and 45 use a holographic grating and a reflection type grating,respectively, the embodiment in FIG. 64 uses a grating 800. Grating 800is, for example, a type of transmission grating, but the output lightfrom grating 800 has a large angle which is in a range of greater thanor equal to 30 degrees and less than or equal to 150 degrees from theinput light to grating 800. For example, in the specific example of FIG.64, the output light from grating 800 is approximately 90 degrees fromthe input light to grating 800. Grating 800 may be made, for example,holographically. The configuration in FIG. 64 is highly practical, sincethe angular dispersion of grating 800 is relatively large within a smallspace.

Moreover, the embodiments in FIGS. 46 and 47 can be applied to theembodiment in FIG. 64. More specifically, quarter wave plate 530, asshown in FIGS. 46 and 47, can be applied to the embodiment in FIG. 64,to cancel the polarization dependence of grating 800.

In FIGS. 44, 45 and 64, mirror 400 could be movable to change thedispersion amount.

For example, FIGS. 65, 66 and 67 correspond, respectively, to FIGS. 44,45 and 64, but show mirror 400 as being movable to change the dispersionamount. For example, in FIGS. 65, 66 and 67, mirror 400 is movable in adirection 810 along the surface of mirror 400 and perpendicular to theangular dispersion direction 402 of VIPA 240.

As should be understood from the above, in various embodiments of thepresent invention, the amount of chromatic dispersion can be changed.Therefore, the VIPA operates in conjunction with other elements (such asa mirror, lens, etc) to provide a tunable chromatic dispersioncompensator.

FIGS. 68 and 69 illustrate a tunable dispersion compensator, accordingto another embodiment of the present invention. Referring now to FIGS.68 and 69, the tunable dispersion compensator comprises avariable-curvature mirror 455 whose curvature changes along a dimensionparallel to the angular dispersion direction of a VIPA 340. Asillustrated in FIG. 70, the variable-curvature mirror 455 comprises, forexample, a flat portion 455 a and a convex portion 455 b, although anunlimited number of other configurations are possible.

As illustrated in FIGS. 68 and 69, the tunable dispersion compensatorcomprises a mechanism for rotating the variable-curvature mirror 455about an axis 456 that is perpendicular to the angular dispersiondirection of VIPA 340. Alternatively, axis 456 can be described as beingperpendicular to a plane which includes the traveling directions of thecollimated output lights for different wavelengths from VIPA 340. Therotation around axis 456 allows differently curved portions of mirror455 to be brought into the focal plane of a focusing lens 352.

FIGS. 68 and 69 both illustrate the pathways of light comprising alonger wavelength 464 of a channel and of the light comprising a shorterwavelength 468 of a channel through the tunable dispersion compensator.However, the light comprising any optical channel comprises light of acontinuum of wavelengths. In FIG. 68, variable-curvature mirror 455 isdisposed so that convex portion 455 b intercepts and reflects the longerwavelength light 464 and the shorter wavelength light 468, whereas, inFIG. 69, the mirror is disposed such that flat portion 455 a interceptsand reflects these lights. As previously described, the convex and flatportions 455 a– 455 b reflect lights 464 and 468 back through lens 352and back into VIPA 340 such that a greater magnitude of chromaticdispersion is produced when the convex portion is disposed in the lightpaths.

Because variable-curvature mirror 455 comprises a plate with axis ofrotation 456 disposed on or within the plate, then, in general, the axisof rotation does not coincide with the center of curvature of anyparticular portion of variable-curvature mirror 455. Therefore, rotationof variable-curvature mirror 455 about axis 456 will generally not bringthe differently curved portions of variable-curvature mirror 455 intothe correct position at or near the focus of lens 352. Therefore, themovement of the mirror between its various positions comprises both arotation of variable-curvature mirror 455 about axis 456 as well as atranslation of axis 456 along a pre-defined path. In FIGS. 68 and 69,axis 456 comprises a rod or pin which is caused to move along a slot ortrack 458 during rotation from the position illustrated in FIG. 68 tothat illustrated in FIG. 69, and vice versa. Alternatively, thetranslation of the axis might be realized by many other mechanisms. Therotation of mirror 455 and translation of axis 456 are simultaneouslycontrolled in the apparatus so as to bring mirror portions comprisingdifferent magnitudes and types of curvature (convex, concave, andplanar) into the path of the light paths as desired. In this fashion,this apparatus in FIGS. 68 and 69 functions as a tunable chromaticdispersion compensator.

FIG. 71 illustrates a tunable dispersion compensator, according toanother embodiment of the present invention. The compensator in FIG. 71comprises similar components to the compensator in FIGS. 68 and 69except that a mirror assembly 558 of the compensator replaces the singlevariable-curvature mirror 455 in FIGS. 68 and 69. Mirror assembly 558comprises a plurality of mirrors 555 a, 555 b, 555 c, etc. all rigidlydisposed approximately equidistantly from a rotational axis 556. Here, a“plurality” of mirrors indicates two or more mirrors or mirror segments.These mirror or mirror segments generally comprise surfaces with avariety of magnitudes and types of curvature (convex, concave, andplanar). The mirrors or mirror segments 555 a, 555 b, 555 c, etc. areattached to a mirror holder 557 centered upon the rotational axis 556.Through rotation of mirror holder 557 about axis 556, differentrespective individual mirrors of the plurality of mirrors 555 a, 555 b,555 c, etc. may be brought into position so as to intercept and reflectlights 464 and 468.

In FIG. 71, mirror holder 557 comprises a cylinder centered uponrotational axis 556. However, any geometric shape or construction may beutilized for mirror holder 557 provided that the distance from thecenter of lens 352 to the surface of the mirror from which lights 464and 468 are reflected is maintained approximately equal to the focaldistance of lens 352. Although the mirrors or mirror segments 555 a, 555b, 555 c, etc. are illustrated as separate non-contiguous mirrorsegments in FIG. 71, these mirrors may also comprise portions of asingle mirror surface of continuously varying curvature.

FIG. 72 is a diagram illustrating a tunable dispersion compensator,according to another embodiment of the present invention. The tunabledispersion compensator in FIG. 72 comprises the same VIPA 340 andfocusing lens 352 components that also comprise the tunable dispersioncompensator in FIGS. 68 and 69, and the tunable dispersion compensatorin FIG. 71. However, in contrast to the tunable dispersion compensatorin FIGS. 68 and 69, and the tunable dispersion compensator in FIG. 71,there is no curved mirror disposed along the focal line of lens 352.Instead, the compensator in FIG. 72 comprises a plurality of mirrors ormirror segments 655 a, 655 b, 655 c, etc. disposed off said focal lineand a rotating flat mirror 602. The mirror or mirror segments 655 a, 655b, 655 c, etc. generally comprise surfaces with a variety of magnitudesand types of curvature (convex, concave, and planar). Although thesemirrors or mirror segments are illustrated as separate non-contiguousmirror segments in FIG. 72, these mirrors may also comprise portions ofa single mirror surface of continuously varying curvature.

In FIG. 72, the rotating flat mirror 602 is disposed along the focalline of lens 352 opposite to VIPA 340 and at an angle to this line so asto fold the paths of lights 464 and 468 onto one or more of the mirrorsor mirror segments 655 a, 655 b, 655 c, etc. Flat mirror 602 rotatesabout an axis 603 that is disposed at the reflective surface flat mirror602 and along the focal line of lens 352. It is to be kept in mind thataxis 603 shown in FIG. 72 is generally not a physical component but onlya geometrical construction. The pathway (not shown) of light of a centerwavelength of an optical channel intercepts mirror 602 at the positionof rotational axis 603.

The plurality of mirrors or mirror segments 655 a, 655 b, 655 c, etc.comprising the tunable dispersion compensator in FIG. 72 are disposedalong a circular arc 606 which represents a portion of a circle centeredat rotational axis 603. The radius of arc 606 is such that the sum ofthe distance from the center of lens 352 to axis 603 plus the distancefrom axis 603 to arc 606 is equal to the focal distance of lens 352.Therefore, lights 464 and 468 are both focused or approximately focusedat points 472 and 474, respectively, wherein both points 472 and 474 arelocated at the surface of one of the mirrors or mirror segments 655 a,655 b, 655 c, etc. Lights 464 and 468 are reflected by one of theplurality of mirrors or mirror segments 655 a, 655 b, 655 c, etc. suchthat each of lights 464 and 468 returns back to the rotating flat mirror602, back through lens 352 and back into VIPA 340.

In the tunable dispersion compensator of FIG. 72, the amount ofchromatic dispersion produced in the optical signal comprising lights464 and 468 may be varied by rotating flat mirror 602 about its centeras shown by the adjustment direction 604. This rotation of the rotatingflat mirror 602 causes controlled movement of the reflection points 472and 474 from one to another of the fixed set of mirrors or mirrorsegments 655 a, 655 b, 655 c, etc. As previously described, themagnitude and direction of the mirror curvature of the particular mirroror mirror segment of the set 655 a, 655 b, 655 c, etc. from which thesignal lights are reflected back to VIPA 340 determines the amount ofchromatic dispersion produced by the apparatus. The signal lights remainfocused at the surface of each mirror or mirror segment 655 a, 655 b,655 c, etc. according to the disposition of these mirrors or mirrorsegments along arc 606. In this fashion, this apparatus functions as atunable chromatic dispersion compensator.

FIG. 73 is a diagram illustrating a tunable dispersion compensator,according to a further embodiment of the present invention. The tunabledispersion compensator in FIG. 73 comprises the same VIPA 340 as thatwithin the tunable dispersion compensators in FIGS. 68, 69, 71 and 72.However, in contrast to these other tunable dispersion compensators, thetunable dispersion compensator in FIG. 73 does not comprise a focusinglens at the output side of VIPA 340. Instead, the tunable dispersioncompensator in FIG. 73 comprises an off-axis parabolic mirror 702 thatperforms the focusing function. Off-axis parabolic mirror 702 isdisposed at the output side of VIPA 340 so as to intercept and reflectthe lights comprising an optical channel that are output by VIPA 340.The paths of two such lights, a light 464 comprising a longer wavelengthand a light 468 comprising a shorter wavelength, are illustrated in FIG.73. Off-axis parabolic mirror 702 is capable of rotation about an axis703 that is disposed along the line at which the light of a centralwavelength (not shown) of an optical channel intercepts parabolic mirror702. It is to be kept in mind that axis 703 shown in FIG. 73 isgenerally not a physical component but only a geometrical construction.

Off-axis parabolic mirror 702 comprises a focus point 705. Through thefocusing power of off-axis parabolic mirror 702, collimated light raysoutput by VIPA 340 and comprising a central wavelength of an opticalchannel (not shown) are brought to a focus at point 705. Light 464comprising a longer wavelength and light 468 comprising a shorterwavelength of said channel are focused at points 472 and 474,respectively, as illustrated in FIG. 73. Points 472 and 474 are disposedat opposite sides of focus point 705, as further illustrated in FIG. 73.During rotation of off-axis parabolic mirror 702 about axis 703, focuspoint 705 and points 472 and 474 move along an arc 706 that represents aportion of a circle centered at axis 703. The tunable dispersioncompensator in FIG. 73 further comprises a plurality of mirrors ormirror segments 755 a, 755 b, 755 c, etc., whose surfaces aretangentially disposed along arc 706. Although the mirrors or mirrorsegments 755 a, 755 b, 755 c, etc. are illustrated as separatenon-contiguous mirror segments in FIG. 73, these mirrors may alsocomprise portions of a single mirror surface of continuously varyingcurvature.

In the tunable dispersion compensator of FIG. 73, the amount ofchromatic dispersion produced in the optical signal comprising thelights 464 and 468 may be varied by rotating off-axis parabolic mirror702 about the axis 703 as shown by the adjustment direction 704. Thisrotation of off-axis parabolic mirror 702 causes controlled movement ofthe reflection points 472 and 474 from one to another of the set ofmirrors or mirror segments 755 a, 755 b, 755 c, etc. The lightscomprising a signal channel are then reflected by one of the mirrors ormirror segments 755 a, 755 b, 755 c, etc. back to off-axis parabolicmirror 702. These lights are then re-collimated by off-axis parabolicmirror 702 and reflected back to VIPA 340. As previously described, themagnitude and type (convex, concave or planar) of the mirror curvatureof the particular mirror or mirror segment of the set 755 a, 755 b, 755c, etc. from which the signal lights are reflected back to the VIPAdetermines the amount of chromatic dispersion produced by the apparatusin FIG. 73. The signal lights remain focused at the surface of eachmirror or mirror segment 755 a, 755 b, 755 c, etc. according to thedisposition of these mirrors or mirror segments along arc 706. In thisfashion, this apparatus functions as a tunable chromatic dispersioncompensator.

As described above, a mirror is used to reflect light back into a VIPA .Thus, a mirror can be referred to as a “light returning device” whichreturns light back to the VIPA. However, the present invention is notlimited to the use of a mirror as a light returning device. For example,a prism (instead of a mirror) can be used as a light returning device toreturn light back to the VIPA. Moreover, various combinations of mirrorsand/or prisms, or lens apparatuses can be used as a light returningdevice to return light back to VIPA.

In various embodiments of the present invention, a lens is used to focuslight from a VIPA to a mirror, and to direct the returning light fromthe mirror back to the VIPA. See, for example, the operation of lens 252in FIG. 13. However, the present invention is not limited to using alens for this purpose. Instead, other types of light directing devicescan be used in place of the lens. For example, a mirror can be used inplace of lens 252 to focus the light from the VIPA, and to direct thereturning light back to the VIPA.

In the above embodiments of the present invention, a VIPA has reflectingfilms to reflect light. For example, FIG. 8 illustrates a VIPA 76 havingreflecting films 122 and 124 to reflect light. However, it is notintended for a VIPA to be limited to the use of “film” to provide areflecting surface. Instead, the VIPA must simply have appropriatereflecting surfaces, and these reflecting surfaces may or may not beformed by “film”.

Further, in the above embodiments of the present invention, a VIPAincludes a transparent glass plate in which multiple reflection occurs.For example, FIG. 8 illustrates a VIPA 76 having a transparent glassplate 120 with reflecting surfaces thereon. However, it is not intendedfor a VIPA to be limited to the use of a glass material, or any type of“plate”, to separate the reflecting surfaces. Instead, the reflectingsurfaces must simply be maintained to be separated from each other bysome type of spacer. For example, the reflecting surfaces of a VIPA canbe separated by “air”, without having a glass plate therebetween.Therefore, the reflecting surfaces can be described as being separatedby a transparent material which is, for example, optical glass or air.

According to the above embodiments of the present invention, anapparatus uses a VIPA to compensate for chromatic dispersion. For thispurpose, the embodiments of the present invention are not intended to belimited to a specific VIPA configuration. Instead, any of the differentVIPA configurations discussed herein, or those disclosed in U.S.application Ser. No. 08/685,362, which is incorporated herein byreference, can be used in an apparatus to compensate for chromaticdispersion. For example, the VIPA may or may not have a radiationwindow, and the reflectances on the various surfaces of the VIPA are notintended to be limited to any specific examples.

The present invention relates to a VIPA dispersion compensator. The term“VIPA dispersion compensator” refers to an apparatus which uses a VIPAto produce chromatic dispersion, such as those described herein. Forexample, the apparatuses in FIGS. 13, 19, 32, 42, 44 and 48(A), amongothers, show a VIPA dispersion compensator.

The various embodiments of a VIPA described herein can also be referredto as a virtually imaged phased array (VIPA) generator.

Although a few preferred embodiments of the present invention have beenshown and described, it would be appreciated by those skilled in the artthat changes may be made in these embodiments without departing from theprinciples and spirit of the invention, the scope of which is defined inthe claims and their equivalents.

1. An apparatus comprising: a virtually imaged phased array (VIPA)generator receiving an input light at a respective wavelength andproducing a corresponding output light traveling from the VIPA generatorin a direction determined by the wavelength of the input light; aplurality of fixed mirrors having different surface curvatures toreflect light; and a rotating mirror rotatable about a rotation axis toreflect the output light traveling from the VIPA generator to arespective fixed mirror of the plurality of fixed mirrors, and toreflect the light reflected by the respective fixed mirror back to theVIPA generator, to thereby provide dispersion compensation to the inputlight.
 2. An apparatus as in claim 1, wherein the plurality of fixedmirrors are separate, non-contiguous mirrors.
 3. An apparatus as inclaim 1, wherein the plurality of fixed mirrors are portions of a singlemirror surface of continuously varying curvature.
 4. An apparatus as inclaim 1, further comprising: a lens focusing the output light travelingfrom the VIPA generator to the rotating mirror, and directing the lightfrom the fixed mirror and reflected by the rotating mirror back to theVIPA generator.
 5. An apparatus comprising: a radiation window; firstand second reflecting surfaces in parallel with each other, the firstreflecting surface allowing substantially no light to be transmittedtherethrough and being in the same plane as the radiation window, thesecond reflecting surface having a reflectivity which causes a portionof light incident thereon to be transmitted therethrough, wherein aninput light at a respective wavelength travels through the radiationwindow and is focused into a line, and the first and second reflectingsurfaces are positioned so that the input light radiates from the lineto be reflected a plurality of times between the first and secondreflecting surfaces and thereby cause a plurality of lights to betransmitted through the second reflecting surface, the plurality oftransmitted lights interfering with each other to produce a collimatedoutput light which travels from the second reflecting surface along adirection determined by the wavelength of the input light, and isthereby spatially distinguishable from an output light formed for aninput light having a different wavelength; a plurality of fixed mirrorshaving different surface curvatures to reflect light; and a rotatingmirror rotatable about a rotation axis to reflect the output lighttraveling from the second reflecting surface to a respective fixedmirror of the plurality of fixed mirrors, and to reflect the lightreflected by the respective fixed mirror back to the second reflectingsurface to pass through the second reflecting surface and undergomultiple reflection between the first and second reflecting surfaces. 6.An apparatus as in claim 5, wherein the plurality of fixed mirrors areseparate, non-contiguous mirrors.
 7. An apparatus as in claim 5, whereinthe plurality of fixed mirrors are portions of a single mirror surfaceof continuously varying curvature.
 8. An apparatus as in claim 5,further comprising: a lens focusing the output light traveling from thesecond reflecting surface to the rotating mirror, and directing thelight from the fixed mirror and reflected by the rotating mirror back tothe second reflecting surface.
 9. An apparatus comprising: a pluralityof fixed mirrors having different surface curvatures to reflect light;and an off-axis parabolic mirror rotatable about a rotation axis toreflect light produced by a virtually imaged phased array (VIPA)generator to a respective fixed mirror of the plurality of fixedmirrors, and to reflect the light reflected by the respective fixedmirror back to the VIPA generator.
 10. An apparatus as in claim 9,wherein the plurality of fixed mirrors are separate, non-contiguousmirrors.
 11. An apparatus as in claim 9, wherein the plurality of fixedmirrors are portions of a single mirror surface of continuously varyingcurvature.
 12. An apparatus comprising: a virtually imaged phased array(VIPA) generator receiving an input light at a respective wavelength andproducing a corresponding output light traveling from the VIPA generatorin a direction determined by the wavelength of the input light; aplurality of fixed mirrors having different surface curvatures toreflect light; and an off-axis parabolic mirror rotatable about arotation axis to reflect the output light traveling from the VIPAgenerator to a respective fixed mirror of the plurality of fixedmirrors, and to reflect the light reflected by the respective fixedmirror back to the VIPA generator, to thereby provide dispersioncompensation to the input light.
 13. An apparatus as in claim 12,wherein the plurality of fixed mirrors are separate, noncontiguousmirrors.
 14. An apparatus as in claim 12, wherein the plurality of fixedmirrors are portions of a single mirror surface of continuously varyingcurvature.
 15. An apparatus comprising: a radiation window; first andsecond reflecting surfaces in parallel with each other, the firstreflecting surface allowing substantially no light to be transmittedtherethrough and being in the same plane as the radiation window, thesecond reflecting surface having a reflectivity which causes a portionof light incident thereon to be transmitted therethrough, wherein aninput light at a respective wavelength travels through the radiationwindow and is focused into a line, and the first and second reflectingsurfaces are positioned so that the input light radiates from the lineto be reflected a plurality of times between the first and secondreflecting surfaces and thereby cause a plurality of lights to betransmitted through the second reflecting surface, the plurality oftransmitted lights interfering with each other to produce a collimatedoutput light which travels from the second reflecting surface along adirection determined by the wavelength of the input light, and isthereby spatially distinguishable from an output light formed for aninput light having a different wavelength; a plurality of fixed mirrorshaving different surface curvatures to reflect light; and an off-axisparabolic mirror rotatable about a rotation axis to reflect the outputlight traveling from the second reflecting surface to a respective fixedmirror of the plurality of fixed mirrors, and to reflect the lightreflected by the respective fixed mirror back to the second reflectingsurface to pass through the second reflecting surface and undergomultiple reflection between the first and second reflecting surfaces.16. An apparatus as in claim 15, wherein the plurality of fixed mirrorsare separate, non-contiguous mirrors.
 17. An apparatus as in claim 15,wherein the plurality of fixed mirrors are portions of a single mirrorsurface of continuously varying curvature.